Agent based modeling of risk sensitivity and decision making on coalitions

ABSTRACT

An agent based model system provides simulation of the influence of environmental variables and tendencies in individual and social decision making relating to the formation of coalitions and ethnic groups. The system is based on improved understandings of human decision making under risk, and incorporates recent theoretical developments and computational tools. The system gives analysts the ability to predict the development of coalitions and ethnic groups, as well as the ability to manage the behavior of individuals in such groups. The model results provide confidence intervals for various possible scenarios in a mix of agents&#39; decision rules and distribution of environmental resources. Applications include management of ethnic groups and violent conditions in unstable nations, the tracking of terrorist organizations, development of coalitions and oligopolies in business, and the modeling and interdiction of criminal organizations.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Nos.60/600,947 filed Aug. 12, 2004, and 60/653,777 filed Feb. 16, 2005, bothtitled AGENT BASED MODELING OF RISK SENSITIVITY AND DECISION MAKING ONCOALITION FORMATION AND DISSOLUTION, which are hereby incorporatedherein by reference in their entirety.

REFERENCE TO A COMPUTER PROGRAM LISTING APPENDIX

A Computer Program Listing Appendix has been submitted on a singlecompact disc (“Copy 1”) and an identical duplicate copy (“Copy 2”) ofthe compact disc. The contents of the compact disc, and its identicalduplicate copy, include twenty (20) ASCII files entitled: Table01.txt(1,696 bytes, created Apr. 7, 2011); Table02.txt (2,691 bytes, createdApr. 7, 2011); Table03.txt (3,736 bytes, created Apr. 7, 2011);Table04.txt (1,212 bytes, created Apr. 7, 2011); Table05.txt (1,235bytes, created Apr. 7, 2011); Table06.txt (415 bytes, created Apr. 7,2011); Table07.txt (932 bytes, created Apr. 7, 2011); Table08.txt (522bytes, created Apr. 7, 2011); Table09.txt (670 bytes, created Apr. 7,2011); Table10.txt (582 bytes, created Apr. 7, 2011); Table11.txt (1,854bytes, created Apr. 7, 2011); Table12.txt (1,851 bytes, created Apr. 7,2011); Table13.txt (2,542 bytes, created Apr. 7, 2011); Table14.txt(1,199 bytes, created Apr. 7, 2011); Table15.txt (184 bytes, createdApr. 7, 2011); Table16.txt (1,805 bytes, created Apr. 7, 2011);Table17.txt (1,316 bytes, created Apr. 7, 2011); Table18.txt (287 bytes,created Apr. 7, 2011); Table 19.txt (1,207 bytes, created Apr. 7, 2011);and Table20.txt (998 bytes, created Apr. 7, 2011). The contents of thisComputer Program Listing Appendix, filed on compact disc, areincorporated herein by reference.

FIELD OF THE INVENTION

The invention relates generally to agent based systems for socialsimulation, and more specifically to methods and systems for agent basedcomputer simulation of environmental variables and tendencies inindividual and social decision making relating to the formation ofcoalitions and ethnic groups. Additionally, the invention relates to amethod and system for deriving a function describing wealth distributionin a population.

BACKGROUND AND SUMMARY

Agent based modeling (ABM) provides a means to analyze individual andsocial behavior, including the formation of coalitions of individuals,their adoption of ethnic and other norms, and the influence of ethnicand other affiliation on coalition members' behavior. Ethnic normsinclude shared beliefs and behaviors such as religious beliefs,dialects, modes of dress, ideology, and cuisine. Those sharing normsform a coalition of individuals who, at least sometimes, act out ofgroup interest rather than short term individual interest. Howcoalitions emerge, when members adopt norms, what motivates individualsto join such groups, and how group membership influences behavior areissues that have become increasingly important to researchers in thesocial sciences. Recent successes in recruiting among terror groups andpotential alliances among these groups represent security concernsregarding dynamics of coalition formation that may lead to socialunrest, collective violence and/or terrorism. Despite revolutionaryrhetoric, rebellions usually are started and led by the well-off.

Human ethnicity involves coalitions of individuals who adhere to normsof behavior. Anthropology is a particularly rich discipline in which toaddress questions of ethnic coalition formation, given the discipline'simportant cross-cultural comparative data on the formation andextinction of ethnic groups under a variety of environmental, social,and historical conditions.

Anthropologists have researched the origin of coalitions and thedevelopment of cultural norms from cultural ecology or human behavioralecology research programs. In these programs, human behavior is seen asan evolutionary outcome of adapting to physical and social environments.Adaptive theories of coalition formation and ethnicity are grounded inrational choice, or bounded rationality theories. The methodsanthropologists use include optimization, game theory, and computersimulation. Decision rules and other formulations are used to simulateand test proposed theories. In all cases, researchers propose that theirmodels are on the whole, adaptive for meeting agents' material needs,even if agents sometimes adopt norms that fail in particularcircumstances.

A drawback to these otherwise useful investigations is that researchersusually evaluate the performance of only one decision rule at a time,and usually over a limited number of iterations. Furthermore,competition for resources via genuinely competing strategies in largepopulations with varying environments usually has not been simulated oranalyzed discretely. The discrete mathematics involved becomesintractable as dimensions or numbers of players increase and the entriesin payoff matrices fluctuate. Instead, researchers commonly summarizethe effects of model parameters with systems of differential equationsand Lyapunov functions to produce more qualitative, global analyses ofstability or convergence, ultimately obscuring coalitions and tending toanalysis of mean behaviors and variances at best.

Competing theoretical claims of coalition and norm formation may beevaluated using computer simulations. These simulations shed light uponthe emergence of ethnic and other affiliation and identification. Tosimulate the complexity of human social interaction, parallel processingcomputer clusters that have the computing power required for modelinglarge numbers of interacting agents may be used. Such simulations maymodel coalition formations and the development of norms under varyingenvironmental conditions relevant to recent theories of ethnic groupformation.

Advantageously, an illustrative ABM stimulation system can be used toevaluate the effects of numerous environmental conditions, simulatelarge populations, and accommodate more than one type of decision rule.The system can be used to evaluate propositions regarding humanrationality and selfishness, develop simulation methodology for modelingcoalition formation, and to advance substantive understanding of justhow and why coalitions form or disperse. The simulation system providesa test bed for competing theories of macro- and micro-behaviors indynamic games by controlling the fluctuations of environmentalconstraints, information constraints, and proximal interactions amongagents. Post-simulation analysis provides, with some degree ofconfidence levels and reproducibility, measures of the model'ssensitivity to parameters in payoff matrices, resource allocations,agent strategies and choices.

The use of such a system may have broad impacts for both education andpolicy. Policy implications involve shedding light on politicalBalkanization and the rise of ethnic identity in the post-Cold Warworld. The understanding of individual and coalition behaviors subjectto fluctuating environmental constraints or controls is a necessaryfirst step for applications in foreign affairs planning and domesticconcerns, such as political action groups, criminal rings, and communityorganizations.

In accordance with the teachings of this disclosure, a method in acomputer system for simulating individual and social behavior, includes:providing for creating a neighborhood of cells; providing for allocatingresources to the cells; providing for assigning an agent to each cell;providing for selecting one of a plurality of decision rules forinteractions between agents; providing for iteratively conductingdiscrete interactions between the agents using the selected decisionrule and allocated resources; and providing for recording interactionoutcome. The one of the plurality of decision rules may include acoordination game.

In accordance with the teachings of this disclosure, a computer-readablemedium having machine-executable code for simulating the interactions ofagents, includes: code for creating a neighborhood of cells; code forallocating resources to the cells; code for assigning an agent to eachcell; code for selecting one of a plurality of decision rules forinteractions between agents; code for iteratively conducting discreteinteractions between the agents using the selected decision rule andallocated resources; and code for recording interaction outcome. The oneof the plurality of decision rules may include a coordination game.

In accordance with the teachings of this disclosure, a system for agentbased modeling, includes a computer and software enabling the computerto use a coordination game model to simulate and analyze theinteractions of individual agents. The computer may include processorsconfigured for parallel processing of the software.

In accordance with the teachings of this disclosure, a method ofestimating the distribution of wealth in a population comprises applyingthe equation W(x)=e^((d+ax+c sin(bx))), where a>bc. A method ofestimating parameters of an expo-sigmoid model of wealth distributioncomprises: creating a data set by taking the natural log of wealth dataof a population; deriving a linear equation in the form of d+ax byperforming an ordinary least squares regression of the data set;performing a fast-fourier transform on the residuals of the ordinaryleast squares regression; picking terms from the fast-fourier transformwhich are statistically significant to form a trigonometric polynomialin the form of f(x)=Σ cos(x)+sin(x); and constructing an equationestimating the wealth distribution of a population by calculating theexponential value of the sum of the linear terms and the trigonometricterms in the form of W(x)=e^((d+ax+Σ cos(x)+sin(x))).

Additional features, which alone or in combination with any otherfeature(s), including those listed above and those listed in the claims,may comprise patentable subject matter and will become apparent to thoseskilled in the art upon consideration of the following detaileddescription of illustrative embodiments exemplifying the best mode ofcarrying out the invention as presently perceived.

BRIEF DESCRIPTION OF THE DRAWINGS

The detailed description particularly refers to the following figures,in which:

FIG. 1 is graph showing an illustrative distribution of resourcesobtained and valued by individuals in a society, in this case wealth,the distribution being a sigmoid function;

FIG. 2 is a graph showing an illustrative distribution of resourcesobtained and valued by individuals in a society, in this case socialrank, the distribution being a expo-sigmoid function;

FIG. 3A is a graph showing an illustrative distribution of resourcesobtained and valued by individuals in a society, specifically wealth inMayan society, the distribution being an expo-sigmoid function;

FIG. 3B is a graph showing an illustrative distribution of resourcesobtained and valued by individuals in a society, specifically wealth inUS society in 2000, the distribution being an expo-sigmoid function;

FIG. 3C is a graph showing an illustrative distribution of the worldpopulation ranked by wealth and risk sensitivity related to thedistribution;

FIG. 4 is a graph showing an illustrative distribution of income amongPalestinian society and risk sensitivity related to the distribution;

FIG. 5 is a table illustrating the join and defect resource payoffs fora type of decision rule, specifically the coordination game;

FIG. 6A is a graph showing an exemplary initial wealth distributiondescribed by the function W(x)=15x−400 sin(π/128x), a simulationiteration 500 wealth distribution described by the functionW(x)=2820+15x−503 sin(π/128x), and classes of individuals and their risksensitivities;

FIGS. 6B and 6C are aligned graphs showing the exemplary initial anditeration 500 distributions of FIG. 6A and the corresponding AP risksensitivity for each rank generation;

FIG. 6D is a table of the observed joins and expected joins for anexemplary simulation among the classes identified in FIG. 6A;

FIG. 6E is a table of the in-class and cross-class coalition joins foran exemplary simulation among the classes identified in FIG. 6A;

FIGS. 6F-6G are graphs illustrating the social (resource) mobility ofsample agents in the exemplary simulation of FIGS. 6A-6E;

FIG. 6H is a graph illustrating the social (resource) mobility of allagents in the exemplary simulation of FIGS. 6A-6E;

FIGS. 7A-7D is a flowchart showing an illustrative ABM simulationalgorithm for modeling a optimal mixed strategy for agent interactionand which may be implemented by software in a computer system;

FIG. 8 is a block diagram showing an illustrative Beowulf clusterparallel processing computer system;

FIG. 9 is a graph illustrating the world wealth distribution (PricePower Parity index, PPP in $US) fitted with an illustrative expo-sigmoidcurve on the original data, the expo-sigmoid curve beingW(x)=e^((−0.438+0.4x+0.399 sin(x))), with all coefficients significantat the p<0.0001 level, R²=0.966;

FIG. 10 is graph showing an illustrative distribution of resourcesobtained and valued by individuals in a society, in this case wealth,the distribution being a sigmoid function;

FIG. 11 is graph showing an illustrative distribution wealth and classesof wealth associated with risk sensitivity;

FIG. 12 is graph showing an illustrative comparison of AP and downsiderisk aversion measures with sigmoid utility, U(x)=x−sin(0.5x), APr(x)=−0.25 sin(0.5x)/(1−0.5 cos(0.5x)), downside risk adversiondra(x)=0.125 cos(0.5x)/(1−0.5 cos(0.5x)), carrying capacity K=2π/0.5=4π;

FIG. 13 is a graph showing an illustrative wealth distribution (laborcost of residence) in a sample of Late Classic Maya households at Copan;

FIG. 14 is a graph showing an illustrative distribution of physicalwealth (land, capital, slaves in the American Colonies, 1774;

FIGS. 15A-15B are graphs showing the entire sample and a sub-sample ofan illustrative distribution of family income in the US, 2000;

FIG. 16 is a graph showing the global distribution of wealth in 1999,and illustrating the basically convex containing sigmoid oscillations incertain sections;

FIG. 17 is a graph of third world wealth distribution and anillustrative fitted curve, Adjusted PPP=0.927*xppp 10−sin(0.793*xppp10), R²=0.784 and parameters statistically significant at well below the0.00001 level;

FIG. 18 is a graph of the wealth distribution for the world's wealthiest54 nations and an illustrative fitted curve, Adjusted PPP=15.654*xppp10−sin(3.966*xppp 10)−116.675, R² of 0.923 and parameters statisticallysignificant at well below the 0.00001 level;

FIG. 19 is a graph of fitted curves predicting world wealth distributionin 1999;

FIG. 20 is a graph of the distribution of wealth among 36 industrializednations in 1987 and the best fitting curve AdjustedPPP=4.18*population−sin(2.14*population), R²=0.965, statisticallysignificant at well below the 0.00001 level;

FIG. 21 is a table illustrating outcome expectations of the illustrativeABM system for strengths of effects relative to different decisionrules;

FIG. 22 is a graph illustrating a simple sigmoid wealth distributionamong agents used to initialize the illustrative ABM system forsimulating nepotism;

FIG. 23 is a graph illustrating the expo-sigmoid wealth distributionamong agents after 1000 iterations of the illustrative ABM systemsimulating nepotism;

FIGS. 24A and 24B are a data table of Palestinian wealth used as inputdata for the initial distribution of resources for an illustrativesimulation of Palestinian society;

FIG. 25 is a scaled and sorted graph of the data drawn from FIGS. 24Aand 24B;

FIGS. 26-32 are graphs illustrating aspects of deriving an expo-sigmoidfunction for the scaled and sorted input data of FIG. 25;

FIGS. 33-37 and 38A-38F are graphs collectively showing an illustrativeinput data set and simulation data output for the illustrative ABMsystem;

FIG. 33 illustrates an initial wealth distribution according to aderived sigmoid distribution W(x)=15x−400 sin(x);

FIG. 34 illustrates the calculated join probability by wealth;

FIG. 35 illustrates the AP measure of risk sensitivity by wealth rank;

FIG. 36 illustrates the wealth mobility (the difference between anagent's lowest rank and highest rank in a simulation) of agentsaccording to wealth rank;

FIG. 37 illustrates the volatility, or the sum of the absolute value ofwealth gains and losses for a simulation; and

FIGS. 38A-38F illustrate the rank history for specific iterations ofgame play for various agents.

DETAILED DESCRIPTION

The illustrative agent based model (“ABM”) system provides simulation ofthe influence of environmental variables and tendencies in individualand social decision making relating to the formation of coalitions andethnic groups. The system is based on improved understandings of humandecision making under risk, and incorporates recent theoreticaldevelopments and computational tools. The system gives analysts theability to predict the development of coalitions and ethnic groups andto adapt policies and actions accordingly, as well as the ability tomanage the behavior of individuals in such groups. The model resultsprovide confidence intervals for various possible scenarios in a mix ofagents' decision rules and distribution of environmental resources.Applications include management of ethnic groups and violent conditionsin unstable nations, the tracking of terrorist organizations,development of coalitions and oligopolies in business, and the modelingand interdiction of criminal organizations.

The distribution of resources obtained and valued by individuals in asociety may be modeled by a sigmoid function, for example as shown inFIG. 1, or a expo-sigmoid function, for example as shown in FIG. 2.Resources include well-being, including, for example, monetary wealth,social status and social worth. Culturally defined wealth is the mostcommon measure of social status and people are often willing to take arisk which may result in a loss or gain of status when the amount ofpossible gain in status from taking the risk exceeds the amount ofpossible loss in status from taking the risk.

As shown in FIG. 1, the people most likely to take a risk to gainresources are located in the rank area(s) of convexity x. Simplesocieties, for example foraging societies, commonly have wealthdistributions that may be described by a sigmoid function, for examplein the form r(x)=−U″(x)/U′(x), where U(x)=αx−sin(βx), α>β>0, and x>2π/β.Complex societies, for example ancient states such as Mayan society, forwhich the wealth distribution is illustrated in FIG. 3A, and modernstates such as the United States, for which the wealth distribution of2000 is illustrated in FIG. 3B, most often have wealth distributions inwhich elites control a majority of the wealth. Such wealth distributionsexhibit an overall exponential distribution of wealth with sigmoidfluctuations around this trend and can be described by an expo-sigmoidfunction, for example in the form W(x)=e^((d+ax+c sin(bx))), where a>bc.Wealth distribution between societies, for example between countries,may also be described by an expo-sigmoid function as illustrated in FIG.3C. FIG. 3C illustrates convexity, thus risk proneness, in areas of thedistribution where countries known for Jihadist recruiting are located,including Iran, Syria, and Pakistan.

One measure of risk proneness and risk adversion well know in the art isthe Arrow-Pratt (“AP”) measure of local risk aversion where U is a twicedifferentiable utility function r(x)=−U″/U′. Negative values correspondto risk proneness and positive values correspond to risk adversion, forexample as shown in FIG. 4, which illustrates income distribution andrisk sensitivity within Palestinian society.

Computer-based or other computationally-based models of individual andsocial behavior include an ABM. The present illustrative ABM system, aflowchart of which is illustrated by FIGS. 7A-7D simulates risksensitive behavior, dynamics of resource flow, and coalition formation,and can identify, for example, individuals (“agents”) vulnerable toterrorist or insurgent recruitment. There is a long history of usingprisoner's dilemma games to model the tradeoffs between joining with, ordefecting against, others. Recent research indicates that another classof game, the coordination game, best models human interactions wherecooperation potentially yields benefits. The illustrative ABM systemdisclosed herein utilizes a form of the coordination game, the join anddefect resource payoffs of which are illustrated in FIG. 5. Measured orotherwise collected data for a society, for example the wealthdistributions of FIGS. 3A-3C can be used by the illustrative ABM systemto evaluate influences of various factors on agent interaction and topredict future or confirm historic individual and social behavior.

As a starting point for analysis to evaluate models and predictindividual and social behavior, the decision rule for agents can be therationally optimal Nash equilibrium strategy that, by definition, offersan individual agent the best long-run payoff, considering what all otheragents are doing. No player may profitably deviate from his strategywhile all others retain their same equilibrium strategies. For example,as illustrated by FIG. 5, where R>T=P>S, or more specifically, R=5, P=3,T=3, and S=0, if Row player agrees to Join, but Column player Defects,Row would receive 0 and Column, 3. However, if both agree to Join (acoalition), then Row receives 5 and Column, 5. The best strategy is toplay Join (thereby forming a coalition) a proportion, p, of the time andDefect (refusing to join a coalition) 1−p of the time. The values of thepayoffs are key influences in inducing coalition formation. For theexample values discussed above, p=⅗. Beginning with this payoffstructure, analysis can be performed on how changes in the payoffs tojoining or defecting influence coalition formation.

The illustrative ABM system may be used to explore sufficiency of therational choice model for explaining and predicting coalition formation.Researchers have noted that wealth inequities influence risksensitivities of agents. Risk sensitivity refers to the likelihood thatan individual will accept an unfair gamble, or will insure againstlosses in a fair gamble. Risk sensitivity alters the payoffs in thecoalition game matrix and therefore the likelihood of Joining orDefecting. People on the verge of entering a higher social status or whoare absolutely destitute are typically willing to take risks whereasthose well ensconced within a class tend to be averse to risks. Suchtendencies could, for instance, explain the rise of militant clerics andtheir recruitment from the very poor in present day Iraq. Anotherexample would be Al Qaeda's and perhaps Hamas' recruitment fromimpoverished populations in south Asia. One of the advantages of theillustrative ABM system is the ability to explore how emergent wealthinequities influence agent risk sensitivity and the consequent formationor dissolution of coalitions, providing a deeper understanding of theirunderlying causes.

ABM simulations typically provide for arming agents with one type ofdecision rule, such as rational maximization of satisfaction. However,researchers have proposed other decision rules based on imitationheuristics that can lead to the formation of ethnic groups. Conformistheuristics involve individuals assessing what behaviors are most commonand imitating those behaviors. Prestige heuristics involve individualsidentifying high status individuals and imitating their behavior.Research to date has focused on demonstrating the effectiveness of aparticular theory or decision rule. The illustrative ABM systemaccommodates more than one type of decision rule, allowing more thoroughtests and comparison of current and competing theories, leading to thedevelopment of more powerful theories and associated ABM systems.

For example, analysis of output data from iterations of the illustrativeABM system demonstrate that the amplitude of resource class boundariesincrease with iterations (time), solidifying class boundaries (FIG. 6Ashowing an initial wealth distribution described by the functionW(x)=15x−400 sin(π/128x) and an iteration 500 wealth distributiondescribed by the function W(x)=2820+15x−503 sin(π/128x)); the agents inconvex wealth distributions form risk prone coalitions (FIGS. 6B-6D);the agents form coalitions across class boundaries (FIG. 6E); and riskprone agents experience greater social mobility (FIG. 6F illustratingthe changing rank over time of agents 1 and 16; FIG. 6G illustratingagents 112 and 128; and FIG. 6H illustrating all agents).

Nearly all established ABM's use a grid upon which researchers buildabstract environments, and over which agents move and communicate withnearest neighbors. The illustrative ABM system provides a scalable gridof cells that may be integrated with available and future geographicinformation system (“GIS”) and other databases. The illustrative ABMsystem also provides for agent communication and movement across thelandscape (cells), for example with vehicular or air transportation, andelectronic or other communication. Additionally or alternatively, theABM system may utilize a conceptual universe relying on a constructother than a grid of cells, thus providing simulation of more complexenvironments and interactions, including geography, nation-states,kinship and the like. For example, the conceptual universe may bedefined as overlapping regions or related functions that best representthe environment being simulated and data being input into the ABMsystem.

Agents can be configured with initial wealth levels, decision rules,sex, age, and other features that model an agent's well being. Detailedhistories of each agent's states (wealth, group affiliation, status,etc.) can be stored and analyzed. As coalitions form, histories ofmembership, group size, and member wealth may also be recorded. Thesesimulation outputs provide the basic data for analyzing what factors(environmental, decision rules, chance) actually lead to the formationof coalitions, alliances of coalitions, and the dissolution ofcoalitions. Simulation results, such as coalition boundaries or wealthdistributions, can be displayed in graphical or other form.

Coalition formation can be based on pair-wise or complex agentinteractions. For example, for pair-wise interaction in the illustrativeABM system, the Moore neighborhood of agents in which each agentinteracts with one of its eight nearest neighbors is utilized.Alternatively, interaction may include non-adjacent or multiple agentinteractions. Coalitions are identified as agents, for example adjacentagents, who play Join during a particular iteration of game play. Inorder to analyze the effects of ethnicity or other factors, simulationscan utilize a decision rule that incorporates ethnic affiliations orother factors appropriate to the analysis. For example, for ethnicaffiliations, a decision rule would enhance cooperation among members ofa common ethnic group and decrease the likelihood of interaction withmembers of other groups.

Referring to FIGS. 7A-7D, algorithm 100 illustrates a flowchart for anillustrative ABM simulation system modeling a optimal mixed strategy foragent interaction and which may be implement, for example, by softwarein a computer system. Typically, ABM software is designed for use onsequential processing desktop computers, and is employed fordemonstrating particular theories of decision-making that may explainthe formation of coalitions and ethnic groups. In order to resolvescientific debates regarding which theories more adequately explainethnic behavior, and in order to provide for more realistic simulations,scientists need more flexible ABM's run on parallel processing computerclusters. For example, FIG. 8 illustrates an illustrative parallelprocessing computer system 200, in this example a Beowulf cluster.

The system 200 may include, for example, a server 202 coupled to acluster of personal computer workstations 204 or other computerprocessors via a local area network (“LAN”) 216 and a LAN switch 206.For controlling and monitoring the system 200, a keyboard, video, andmouse (“KVM”) switch 214 may be used to selectively couple a keyboard208, a monitor 210, and an input device 212 to the server 202 and theworkstations 204. Additionally, the server 202 or another element of thesystem 200 may be coupled to the Internet 218 or another public switchedcommunications network or private network, for example a fiber-opticnetwork, for interconnectivity with data sources, data storage,large-scale memory, and/or other processors.

A variety of Java-based software frameworks exist for implementing ABMsimulations, including Swarm, Ascape, MASON, and Cybele. All of theseframeworks have rich sets of resources for constructing complex agentsand environments, visualizing their interactions, and monitoringstatistics of interest. The Cybele architecture supports a distributedcomputing environment and can be used to overcome the limitations ofsingle-processor environments that have reduced the scope of ABMsimulations to relatively small populations impacted by fewenvironmental and individual factors. More realistic simulations involvecomplex environments and populations of thousands of agents. To createsuch simulations, the illustrative ABM system may be implemented on aBeowulf cluster such as that illustrated in FIG. 8. Such a computingenvironment permits running concurrent simulations to evaluate a widerange of parameter sets.

The algorithm 100 may be configured to be implemented on a parallelprocessing system such as the system 200 of FIG. 8, or may be configuredto be executed by a sequential processing computer system or othercomputational devices known in the art. The algorithm 100 begins at step102 and is executed by the processor(s) (hereinafter “processor”) of thesystem 200. At step 104, the processor defines the neighborhood forsimulation game play. For example, the neighborhood may be defined as aMoore neighborhood universe in which a two-dimensional grid of n×n cellsare mapped as onto a sphere, giving each cell a neighborhood of eightadjacent (bordering) neighbors (N, NE, E, SE, S, SW, W, NW). Theneighborhood universe may be alternatively geometrically ornon-geometrically defined and neighborhoods may also be defined toinclude non-adjacent cells in the neighborhood of a particular cell.

At step 106, an allocation of resources (as defined above) is selected,and allocated to the cells. If selected, at step 108 the processorallocates resources uniformly across the universe, for example assigning(total resources)/n² to each cell for an n×n grid. Alternatively, ifselected, at step 110 the processor allocates resources non-uniformly,for example, as a sigmoidal or expo-sigmoidal function of the formdescribed above. The function may be derived from Fourier signatureanalysis fit of residuals from a linear regression to collected data,or, for an expo-sigmoidal function, from a logarithm of collected data.Deriving the sigmoidals function is described in further detail below.

At step 112, the processor assigns agents to cells, for example, oneagent to each cell for a total of n² agents for an n×n grid.Alternatively, agents may be non-uniformly distributed to cells, forexample according to a geographic or other data set or function.

At step 114, the join strategy for the coordination game play isselected for determination of the join probability and resources payofffor an agent joining or defecting or otherwise interacting with otheragents. At step 116, for the rational choice variation, each agent isgiven the optimal mixed strategy with a choice biased according to theoptimal mixed strategy of joining=⅗ and defecting=⅖. For eachinteraction of agents in each iteration of game play, the choice ofjoining or defecting is made according to a random choice biasedaccording to the joining and defecting probabilities.

At step 120, the processor alters the probability of joining anddefecting based on an AP risk posture. At step 122, the processor sortsagents according to resources, for example, wealth or status asillustrated in FIGS. 3A-3C. At step 124, the processor usesquasi-Fourier Analysis for an initial AP calculation, for example, asigmoidal or expo-sigmoidal function that may be derived from Fourieranalysis fit of residues from a linear regression to collected data, or,for an expo-sigmoidal function from a logarithm of collected data. Forexample, as illustrated by FIG. 9. Alternatively, the processor mayrefit sorted agent resource distribution after each iteration ofinteractions by all agents. At step 126, the processor calculates the APrisk posture based on the sigmoid or expo-sigmoid function, for exampleas illustrated in FIG. 12. If AP is ≧0, then the processor assigns agentjoin probability JP=(⅗)*(1−AP/maximum AP), JP=⅗+(AP/minimum AP)*(1−⅗).

At step 128, for a conformist-biased variant an individual agent'schoice to join or defect is based on the dominate strategy in theagent's neighborhood during the last iteration of game play. At step130, the processor calculates the dominate strategy in the agent'sneighborhood, for example choice of joining or defecting made by theagent and the agent's eight neighbors for a two-dimensional griduniverse. At step 132, the calculated dominate strategy is assigned tothe agent for the present iteration of game play.

At step 134, for a prestige-biased variant an individual agent'sstrategy is influenced by the neighbor having the greatest totalresources, for example the greatest wealth or status. At step 136, theprocessor assigns the strategy of the neighbor with the greatestresources to the individual agent.

At step 138, game play is initiated by selecting the number M ofiterations of discrete interaction by each individual agent. At step140, the processor selects the first agent for game play. At step 142,the processor randomly selects a neighbor for interaction.Alternatively, the selection of an agent for interaction may be biasedor determined according to other functions or data sets as furtherdescribed herein, for example an agent may interact with more than oneagent, may interact with a non-neighboring agent, or may be biasedtoward Joining kin.

At step 144, the processor determines whether the neighbor selected forinteraction is busy. If so, execution of the algorithm 100 returns tostep 142, else step 146 is executed. At step 146, the processordetermines whether the neighbor selected for interaction is resourcedepleted and not available for a game play. If so, execution of thealgorithm 100 continues at step 148, else execution continues at step150. At step 148, the processor determines whether an alternativeneighbor is available for game play. If so, execution of the algorithm100 returns to step 142 in order to select another neighbor forinteraction. If at step 148 the processor determines that no otherneighbor is available for interaction then at step 149 the processorassigns no interaction to the agent for the present iteration of gameplay and execution of the algorithm 100 continues at step 152.

If at step 146 the processor determined that the neighbor selected forinteraction is not resource depleted, then at step 150, the processorchooses a join or defect strategy according to steps 114-136 discussedabove. At step 152, the processor assigns resources payoffs between theinteracting agents according to the chosen join or defect strategy andthe resulting resource payoffs. At step 154, the processor recordsagents resources and choices resulting from the present interactions. Atstep 156 the processor determines whether there are agents remaining tointeract for the present iteration of game play. If so, at step 158 theprocessor indexes to the next agent available for game play and theexecution of the algorithm 100 returns to step 142. If at step 156, theprocessor determines that there are no agents remaining for interactionthen execution of the algorithm 100 continues at step 160.

At step 160 the processor displays or otherwise stores or tabulates thestrategy for cells selected in the current iteration of game play.Additionally or alternatively, other results or analysis of results mayalso be displayed, stored, or otherwise tabulated for the most recentiteration of game play. At step 162, the processor records the joincount for the just completed iteration. Additionally, the processor mayalso record the defect count and other aspects of the interactions andresulting payoffs for the iteration. At step 164, the processor maycalculate and display a graph of the universal join relative frequencyUJRF=(total join count)/n².

At step 166, the processor detects coalition neighborhoods. For example,a neighborhood may be labeled as a coalition neighborhood if the joincount amongst the nine agents in a neighborhood exceeds the UJRF. Atstep 168, the processor detects adjacent coalition neighborhoods, forexample, neighborhoods are adjacent if their intersecting cells are notempty. A coalition may be, for example, the union of adjacent coalitionneighborhoods and the size of the coalitions equal to the number ofcells in the union of adjacent coalition neighborhoods.

At step 170, the processor displays the coalitions. At step 172, theprocessor records additional results of the just completed iteration andthe result in coalitions, for example the coalition size, join count,total resources, and sole locations. At step 174, the processor collectsa metabolic tax from each agent's resources. At step 176, the processorremoves resource depleted agents from game play. At step 178, theprocessor determines whether the number of iterations M has beenreached. If not, at step 180 the processor index game play to the nextduration and the algorithm 100 returns to step 140. If at step 178 theprocessor determines that game play is complete, then the algorithm 100continues at step 182 for post simulation analysis. Steps 182-188include illustrative post simulation analysis; however such analysis mayinclude other analyses including that disclosed elsewhere herein.

At step 182, the processor sorts and displays coalitions by size. Atstep 184, the processor chooses signature analysis to detect dominantcyclical patterns. Fourier analysis is used to detect these cycles andtheir dominant periods. At step 186, the processor detects fractalboundary signatures if any exist. This is done by sorting measures ofcoalition size or clustering by their frequencies. These data arelogged, and inspected for scale-free power distributions. At step 188,the processor calculates and displays a plot universe join history.Coalitions are represented by color-coding agents according to whichcoalitions they belong. At step 190, algorithm 100 is complete.

Tables 1-20 (set forth in the Computer Program Listing Appendix) listillustrative code implementing the illustrative algorithm 100.Specifically, the code illustrated by Tables 1-20 is C++ source code andheader modules suitable for execution by a sequential processingcomputer. Illustrative input and output data associated with theillustrative software of Tables 1-20 is illustrated in FIGS. 22-38.

Additionally or alternatively, the illustrative algorithm 100 or otheralgorithms within the scope of the invention may be parallelized forexecution, for example, by the parallel processing computer system 200.For example, the server 202 may execute code for regulatingcommunications among the workstations 204 and for compiling andanalyzing the data outputs from the workstations 204. The individualworkstations 204 may each execute code for similar or same repetitivetasks simultaneously.

A parallelized software system architectural design may be viewed as atop-down arrangement of independent units, capable of acting in parallelwith each other on separate cluster nodes of the system 200, or insynchronization with high-speed communication links. Each unit isidentified as an independent unit. Initially, some sequential processingof “macro-independents” is required to effect system operation and isreflected in the following listed order: Control Panel, GoverningAuthority, Agent Mind and Control, Data Collection, Real-Time Data Maps,and Data Mining.

The independent units, themselves, may be further divided into parallelprocesses to take advantage of idle nodes. Should, the initial, staticmapping of processes onto nodes become less than optimal, dynamicmapping of process onto idle nodes may be allowed to enhance thecomputational throughput through a “re-docking process” in the controlpanel. The following is intended to provide a description of eachmacro-independent and is only one alternative guideline for actualimplementation onto a cluster. The actual cluster hardware configurationand communication capabilities will impose unforeseen adaptations onindividual clusters. One advantage of the illustrative algorithm is thatit provides for such modifications through task separation andindependence.

The macro-independent Control Panel enables initial and real-timecontrol of many aspects of algorithm, system and hardware. It is thedashboard/human-computer-interface for the modeler. Parameter controlsmay provide slider-type interfaces for continuous variation of inputparameters/variables and multi-state switch interfaces for multiplediscrete state selections, for example mind control selection for agentsand interactive coalition labeling. The Control Panel enables the userto manage governing authority, control real-time population profilegraphs, pause and review, and re-construct with docking authority ascausal/plausability analysis.

The macro-independent Agent Mind and Control allows for and engendersemergent behaviors in agents and coalitions. In short, with thisalgorithm, agents become more than “ants” searching a landscape forresources. They behave. The agent categorization and mind control mayinclude the herein discussed interaction strategies, including rationalchoice, AP risk analysis and join probability adaptation,conformist-bias, and prestige-bias; as well as other aspects such asprivacy/visibility to other agents and access to other agent history andglobal data.

The macro-independent Governing Authority proscribes agent actions and“daily” lives. For example, conducting game play, including agentselection for play, pairing or grouping for interaction, inter-agentcommunication protocols, and agent interaction strategies. The GoverningAuthority may also provide census authority as data reporting, initialresource allocation, resource variation during game play, and gamerewards.

The macro-independent Data Collection provides data recording withcarefully controlled access. Data may be collected and recorded inmemory or other known data storage devices, or transmitted for recordingand analysis by another system. Data that is collected may include agentplay histories, mindsets, wealth, status, ranks, coalition memberships,fitness and other measures of well-being and status and game playresults, population global profiles and analysis, and control panelsettings.

The macro-independents Real-time Data Maps and Data Mining provide dataanalysis, mapping and visualization for simulation monitoring andcontrol. Real-time data analysis may include global mappings such asresource allocation, resource depletions, agent life cycles, and agentranking/prestige; granular mappings such as neighborhoods and coalitionboundary mappings; real-time data and analysis displays; and systemmonitoring such as throughput/bandwidth and feedback response timemonitor.

Development of Models Using Expo-Sigmoid Utility. As discussed above,while basically sigmoid wealth distributions are common, complexsocieties in which elites control a majority of the wealth exhibit anoverall exponential distribution of wealth with sigmoid fluctuationsaround this trend. For example, FIGS. 3C and 9 illustrate world wealthdistribution. For FIG. 9, the wealth distribution (Price Power Parityindex, PPP in $US) is fitted with an expo-sigmoid curve on the originaldata. Wealth distribution equation for the scaled data isW(x)=e^((−0.438+0.4x+0.399 sin(x))), with all coefficients significantat the p<0.0001 level, R²=0.966.

The general form of the exponential extension of the original sigmoidalformulation which models such distributions is W(x)=e^(d+ax+c sin(bx)),a>bc. W is used for this function to avoid confusion with traditionalutility functions, although a systematic relationship between W and theU of utility functions holds. The parameter, a, is the same as in thesigmoid function discussed herein. Parameter c is an additionalparameter that allows for amplitude in the sigmoid fluctuation aroundthe underlying exponential distribution. The constant, d, is the naturallog of the y-intercept of the expo-sigmoid distribution. A scalingmethod that sets b=1, providing a simple periodicity of 2π.

This expo-sigmoid function is a robust model of the overall distributionof wealth in a complex economy, allows calculation of risk sensitivitymeasures for all individuals, and facilitates deeper mathematicalanalyses. The function's parameters are easily estimated with non-linearregression techniques that provide extremely close (over 95% varianceexplained) statistical fits to data. Furthermore, the underlyingfunctionality allows identification of risk prone individuals with risksensitivity measures, even in the relatively wealthier segments of asociety where sigmoid fluctuations are less apparent graphically.

Application of Fast-Fourier Transforms for estimating parameters of theexpo-sigmoid model provides for the more difficult fitting of a curve todata in the upper range of a wealth distribution. Fast-FourierTransforms may be used to identify the dominant and statisticallysignificant periods of the sigmoid fluctuation around the exponentialpattern in wealth distribution data. An illustrative method is asfollows and also can be used for modeling measures of other resourcesand other data:

-   -   1. Take the natural log of wealth data to remove exponential        effects.    -   2. Perform an ordinary least squares (“OLS”) regression to        extract the linear effects from the data, resulting in an        equation d+ax.    -   3. Perform a Fast Fourier Transform on the residuals of the OLS.        This produces a list of complex numbers, the real part        corresponding to the coefficient of a cosine of x term, and the        imaginary part corresponding to the coefficient of a sine of x        term. Pick out the terms that are statistically significant from        the Fast Fourier Transform analysis and this results in a        trigonometric polynomial of the form: f=Σ cos(x)+sin(x).    -   4. Construct an argument for the expo-sigmoid function by adding        the sine and cosine terms from the Fast Fourier Transform        analysis, to linear and constant terms from the OLS, resulting        in: W(x)=e^(d+ax+Σ cos(x)+sin(x)).

Development of Models Using Sigmoid Utility. Key issues in the nature ofdecision making include whether or not individuals maximize utility, howmuch culture conditions values, and how and to what extent risk anduncertainty play a role in decision making. A connection betweenenvironmental constraints and decision making can be made by modelingthe implications of potential environmental constraints on risksensitivity and the value of goods. A sigmoid (S-shaped) utilityformulation exhibits both the ecological and social dimensions of valueunder risk. The general applicability of the sigmoid model is indicatedby widespread cross-cultural support in Industrial and non-Industrialeconomies, examples of which are given below, for example, data on thecollapse of an ancient Mayan city-state in Central America, the AmericanRevolution, and contemporary voting behavior in the United States.

Researchers commonly assume that marginal value decreases as one obtainsmore of a good, implying that value functions are concave and thatdecision makers are risk averse. The term material wealth is used hereinto refer to actual goods, capital, or social rank, and the moregenerally accepted terms utility and value to refer to the satisfactionone derives from possession of material wealth or social rank. Despitethe general acceptance of and empirical support for risk aversion, it isnot universal. Others have argued that a decision maker's sensitivity torisk would vary with material wealth and status. At some wealth levelsdecision makers will be risk prone, preferring a risky gamble with a lowexpected value but having the possibility of winning a large reward overmore assured outcomes. Such a gamble can be considered an unfair bet. Atdifferent wealth levels decision makers would avoid such risks,preferring more assured outcomes with lower expected values thanprobabilistic outcomes.

Milton Friedman and Leonard Savage posited a sigmoid utility curve withalternating concave and convex portions to model such behavior. In thesigmoid model illustrated in FIG. 10, an individual at wealth w in theconvex region of the curve offered an even chance of either increasingor decreasing wealth by p, would take the gamble because the increase inutility, b, is greater than the potential loss of utility, a. Aslabeled, a risk prone individual would have a tendency to take chancesto gain wealth to move upward in social rank. Such a person would inother words be willing to pay a cost, or invest, on a particularly riskyventure. An individual whose material wealth lies at pointy in theconcave region of the curve would be risk averse and refuse the samegamble (refuse to pay a cost) because the most possibly gained would bed, which is less than the potential loss c. Friedman and Savagesuggested that those well established within a particular social stratumwould be risk averse and have concave utility. Such decision makerswould be reluctant to risk losing too much wealth and sliding back to alower social status. In contrast, others on the boundary of a wealthstratum would have convex utility and would take a risk for a potentialgain that would move them to a higher social status. The consequence oflosing for those on the convex portion of the utility curve would be noworse than dropping back to the status they occupied before the gamble.

Full exploration of this proposal requires the specification of asigmoid functional form that can be tested against data and used forfurther analytical research. A sigmoid utility function should exhibitchanges in concavity and be monotonically increasing, and constants insuch a function should directly correlate with observed social factors.The following illustrative sigmoidal analytical function meets thesecriteria: U(x)=αx−sin(βx), where x is the objective measure of materialwealth or social status, α>0 is the cultural emphasis on status, β>2πβis the quasi-period delineating distinctive social strata (classes) withrespect to material wealth, and α>β>0. With the restriction, α>β>0, U(x)is monotonically increasing with respect to material wealth.

The implications of sigmoid utility can be explored analytically byconsidering standard measures of risk sensitivity such as AP anddownside risk aversion. The AP measure of absolute risk aversion,r(x)=−U″(x)/U′(x)=β² sin(βx)/(α−β cos(βx)), as illustrated by FIG. 12.Since the first derivative of the sigmoid utility function is positive,U″(x)<0 implies risk aversion (concave) and U″(x)>0 (convex) impliesrisk proneness. An inflection point of the utility function, U″(x₀)=0,implies risk neutrality at x₀. The zeros of AP risk aversion are at theinflection points, x=nπ/β, n=0, 1, 2, . . . .

The AP measure yields negative (risk prone) values in convex andpositive (risk averse) values in concave sections of the curve.Socio-economic classes would be defined in the step-like areas betweenthe inflection points of steep sections of the curve x=2nπ/β, n=0, 1, 2,. . . as illustrated in FIG. 11. While sigmoid utility is not periodic,divisions between classes are delineated by quasi-periods, 2nπ/β,(2n+1)π/β. FIGS. 10 and 11 show one quasi-period of the sigmoid utilityfunction.

The AP measure is limited because the underlying probabilitydistribution for losses and gains assumed in that model is symmetrical.Some decision makers are willing to accept a gamble with a lowerexpected value, provided that the gamble provides an asymmetricallydistributed probability of loss and gain in which there is a negligibleprobability of a large loss and a high probability of a small loss,offset by a small probability of a very large gain. This observation hasbeen made in detailed studies of industrial firms, pig-raisingoperations, and in Indian peasant agriculture. Such preferences imply aprobability distribution skewed to the right toward higher gains. Thisbehavior indicates sensitivity to the third moment, or skewness, of aprobability distribution. The third derivative of a utility functionwill be positive for a person with downside risk aversion.

Researchers have proposed various measures of downside risk aversionbased on the third moment and have derived a measure analogous to theAP. However, this measure provides negative values for downside riskaversion. For consistent comparison with the AP measure, the measure ofdownside risk aversion must be negated as follows:dra(x)=−U′″(x)/U′(x)=−βb³ sin(βx)/(α−β cos(βx)).

FIG. 12 illustrates that having a convex or concave utility functiondoes not necessarily imply either downside risk averse or risk pronebehavior. Those who are on or straddle a class boundary (quasi-period),are predictably downside risk prone (dra(x)>0), willing to take thegamble of falling back into their class for a long chance at raisingtheir social status. Individuals on or straddling the mid-point of aclass are predictably downside risk averse (dra(x)<0), fearing theirloss of social status. These social insights underscore the importanceof knowing more moments of the underlying probability distributionsassociated with the losses and gains a decision maker is consideringwhen confronting a gamble. Class boundaries are pivotal for theimplications of behavior under risk, as described in the examples below.

Prospect theory and sigmoid utility. AP and downside risk measures aregrounded in standard neoclassical economic theory. Prospect theory wasdeveloped to explain paradoxes (e.g. Allais' Paradox, framing effects,loss aversion) unexplained by traditional utility theory. Research hasestablished how people systematically deviate from rationalexpectations, and can be used to make theories of decision making morerealistic. However, neither prospect theory nor any other alternativeprovides a coherent explanatory framework that can replace utilitytheory, despite its limitations. Therefore, a more fruitful approach isto enrich utility theory by incorporating insights from prospect theory.Two aspects of prospect theory, loss aversion and the framing effect,are relevant to this application of sigmoid utility.

People exhibit a stronger disutility from losses than utility fromgains. It has been proposed that the carrier of utility is not a level awealth, but changes from one's wealth. A sigmoid utility curve, convexfor losses and concave for gains, with the slope of the convex curvegreater than the concave section, can be used to model such changes.Despite the evidence in support of this model, the three examples belowdemonstrate where individuals arguably were concerned with gains, notlosses, and where they exhibited convex utility preferences.

People are strongly influenced by the way problems are framed. Peoplebehave differently if they are considering a payoff as a loss ratherthan a gain. In a now famous experiment people were asked whether or notthey would adopt a treatment, providing information on its success rateand its failure rate. People presented with the treatment's success ratesaid yes, and those presented with its failure rate said no. Thefindings of prospect theorists indicate that people's regard for theirprospects and emphasis on disutility are important contextual factorsthat condition otherwise rational decisions, and these considerationswill be incorporated in the examples below.

Empirical evidence of sigmoid utility. Economic anthropologists, usingan implied AP model of risk sensitivity, have independently corroboratedFriedman and Savage's sigmoid utility model. For instance upper middleclass Maya peasants in Zinacantan, Mexico who anticipate moving into ahigher social class are more likely to try new risky agriculturaltechniques and take the chance for increased productivity. Researchamong peasants around the world confirms these observations. Risksensitivity among four ethnic groups in three societies: indigenousMapuche Indians in Chile, their mainstream Chilean neighbors known asHuinca, the agro-pastoral Sangu of Tanzania, and University ofCalifornia undergraduates, are measured with experiments in which theyoffer subjects different lotteries with real cash awards. Theimpoverished indigenous populations exhibit risk proneness, whileeconomically middle class Huinca and American undergraduates aretypically risk averse.

An empirical study of risk sensitivity among goat and alpaca herders inAndean Peru also fits Friedman and Savage's sigmoid model. This studymeasures the distribution of herd sizes in two traditional Aymara Indiancommunities and assesses their individual risk sensitivityexperimentally. Very poor and wealthy herders exhibit a relativeattraction to risk, and middle-wealth herders demonstrate strong riskaversion, indicating that risk sensitivity alternates with quantitativedifferences in wealth as predicted by a sigmoid model.

The illustrative ABM system can be configured to incorporate Friedmanand Savage's sigmoid utility into a model describing the step-likedistribution of reproductive fitness enhancing goods in non-Westernpastoral and foraging groups, as well as among non-human primates.Biologists have noted similar risk sensitivities among insects, birdsand mammals and other studies of human societies and animal groupsdetail resource distributions and reproductive payoffs that appear to berelated in a sigmoid fashion. In light of these varied studies, sigmoidmodels of utility appear to provide analytical insights into risksensitivity in a wide variety of contexts.

Carrying capacity, finite material wealth, and utility. There aresystematic relationships between utility, resource distribution,environmental structure, and competition. Researchers have recognizedthat there is a finite amount of material wealth in the world at any onetime, even if technological developments enable future increases. Thespecification, at any one time, is important since material wealth canexpand through time due to technological improvements in efficiency.However, during any one period when an individual must make a decision,wealth will be finite. Furthermore, there is a growing awareness thateconomies are constrained by these biotic limits of the environment.

Resources, and therefore material wealth, are finite at any one time.Value is potentially finite as well. Utility functions researchers useto model value are typically linear, concave, or sigmoid. If thematerial wealth upon which value (modeled with these functions) is basedis finite, then the maximum value these functions could attain at anyone time would likewise be finite. The crash of “overvalued” stocks,such as the recent drop in internet stocks and “bubbles” such as thefamous tulip bubble in early 17th century Netherlands indicate thatvalue is neither wholly subjective, nor unconstrained. Limits appear toexist even for subjectively valued tulip bulbs.

Anthropologists who have measured value as finite units of materialwealth, such as energy and time, have been tremendously successful inexplaining behavior in a wide range of environmental and economicsettings. Using optimal foraging methods, researchers have explained thediets of hunters and gatherers in Paraguay, Australia, and the PeruvianAmazon, optimal group size among the Bari fishermen of Venezuela, Eskimoand Indonesian whale hunters, herd management among traditionalpastoralists such as the Aymara of Andean South America and Africancattle herders, and the dispersion of fields among peasant QuechuaIndians of the Peruvian Andes and Medieval English serfs.

Empirical evidence clearly shows that material wealth is never equallydistributed. Inequality in resource distribution is obvious in thecompetition over key fruit sources among non-human primates or thepervasive competition over mates and resources across all humansocieties. In summary, material wealth is finite, the value people canderive from such wealth is potentially finite, and wealth differentialsexist. These implications necessarily imply sigmoid utility functions asdemonstrated by considering the expected behavior at the upper and lowerlimits of material wealth.

Sigmoid utility, risk sensitivity, and the limited distribution ofresources. The upper limit on material wealth and utility at any time inthe illustrative sigmoid model is analogous to the concept of carryingcapacity in ecology. Carrying capacity is a theoretical limit onpopulation growth based on the distribution of limited resources. It isalso a practical limit that empirically exists. An example is thecollapse of human population on Easter Island as its indigenousPolynesian inhabitants over-extended use of agricultural land on thisremote Pacific Island. Prehistoric Anasazi towns of the southwestern USsimilarly failed during the 13th century due to the combined effects ofprolonged drought and erosion caused by traditional agriculture. The endresult was the collapse of Anasazi food production and politicalstructure into warfare and cannibalism. In the illustrative sigmoidmodel, material wealth is finite and the traditional symbol for carryingcapacity, K, is used to designate it (FIG. 11). K marks the end of thelast quasi-period. Given the existence of this limit, economiccompetition becomes zero sum at any one time, providing the basis forsigmoid utility based on material wealth distributions. The zero-sumnature of this competition implies a dynamic exchange of valuethroughout a society as the gains won by some are the losses felt byothers.

With finite wealth capacity, consider the dilemma of those near thislimit of material wealth who have little or nothing to gain and who canloose a great deal. Intuitively, these wealthiest individuals should berisk averse (barring the possibility of further large gains in wealth),especially to large losses. Both AP and downside risk aversion measuresare consistent with this explanation (FIG. 12). At the other end of thespectrum, consider the poorest who have almost no material wealth atall. Any of these individuals immediately above a point of absolutedestitution have almost nothing left to lose and everything to gain ifthey must consider only symmetrical gambles. If presented with gamblespositively skewed toward high rewards, they would be predictablydownside risk averse. Therefore, the utility function will necessarilybe convex near zero material wealth (FIG. 11) with positive downsiderisk aversion (FIG. 12).

Standard measures of risk sensitivity yield reasonable and clearexpectations for behavior in the illustrative sigmoid model, allowingprediction of which individuals are most likely to take chances. Sincethe illustrative sigmoid model is justified in reference to resourcelimitations, a full examination of the illustrative sigmoid model'sexplanatory potential requires a long-term diachronic study ofenvironmental limitation and growing class-like wealth differentials.The long archaeological record of the rise and fall of Mayancivilization provides an ideal opportunity to examine such long-termeffects. Also instructive is a clear case of historical economic growthfollowed by artificially imposed limits to explain revolutionarybehavior in the American colonies, and the implications of sigmoidutility for voting among today's American citizens.

The Collapse of Mayan Copän. Mayan civilization flourished from about AD400 to 850 in present day Central America. At its height, Mayancivilization existed as a number of independent city-states ruled bykings, and managed by lesser nobles and bureaucrats. Maize agriculturesustained the population and served as a basis for the taxation ofpeasants. This civilization, with its pyramids, scribes, astronomers anddense populations, collapsed during the 9th Century AD. Research at theMayan city of Copän provides a detailed record of this collapse.

Copän is situated near the mountainous border of present-day Hondurasand Guatemala. The environment today, like in the past, is heavilycircumscribed. Fertile arable land exists only in small pocketsscattered along major river valleys. A small population of Mayan farmersbegan clearing land in the Copän valley pocket by 3600 BC. Populationgrew rapidly from AD 400 to 750, reaching an estimated peak near 28,000people. Traditional Central American (Mayan and Aztec) agricultureinvolves cutting and burning field plots every few years. This burningproduces severe sheet erosion in the mountainous and wet environments ofCentral America. Archaeologists studying pollen samples from ancientbogs near Copän have charted environmental changes through time, andhave found that severe erosion accompanied rapid population growth inthe Copän valley.

Commensurate with the growth of Mayar population was increasedcompetition over resources and the development of a distinct socialhierarchy with the following classes: the royal family, representativesof elite noble lineages landed courtiers, and commoners. This socialhierarchy was directly related to the material wealth invested in Mayanhouseholds as measured by number of person-hours required forconstruction. As FIG. 13 indicates, Mayan society had class-likedivisions of wealth corresponding to the hierarchy archaeologists infer(A linear regression of labor cost on rank exhibited clear periodicityin the residuals, necessitating a sigmoid model. Using methods describedherein, a sigmoid curve is fit to the Copän data, providing a close(R²=0.888) and highly significant fit.).

These step-like sigmoid differentials became the fault-lines along whichCopän collapsed during the 9th Century AD. By AD 750, carrying capacityhad been reached and breached, as evidenced by widespread malnutrition,first among Copän's heavily taxed peasantry, and eventually throughoutall classes of Mayan society. Through careful dating, archaeologistsdocument an economic expansion previous to this time (legitimating theuse of a sigmoid curve over gains), environmental degradation, andsubsequent social instability. Using hieroglyphic texts preserved in thestones of Copän structures, archaeologists have pieced together thehistory of an elite rebellion against Copän's king. However, theirdescription lacks a theoretical expectation of an elite rebellion. Whywouldn't the elite have continued to support their king? Why was it nota peasant revolt? The illustrative sigmoid model suggests an answer.

Research documents the economic and political rise of elite nobles andrural political figures shortly before the king's overthrow. Based onMayan texts and the dates of their monuments, archaeologists know thatCopän's last king, Yax Pasah, was deposed between AD 820 and 822. Theburning and defacement of the king's main complex indicates that thecoup was violent. While the Copän monarchy disappeared, its lessernobles and commoners did not. Archaeologists now know that a truncatedMayan hierarchy continued to exist for another 150 years, and thatpopulation decline was gradual. It appears that after the elite nobles'successful overthrow of the king, they continued to rule, and they viedfor power among themselves for at least several generations. Referringback to FIG. 13, an illustrative sigmoid model predicts that preciselythese nobles (approximately rank 40) would have been in a position torebel, owing to the extreme convexity of Mayan wealth distribution intheir position, which would have given them tremendous potential gains.

With the elimination of the king's household, the ruling elites formedan oligarchy, but it was one circumscribed by the limits of agriculturalproduction in the Copan valley. Their gains were eventually lost due tothe continuation of severe erosion that began in the 8th century, In theend, the Copan Maya were doomed, and the valley was effectivelyabandoned by the 13th Century, only regaining its population numbers(with a concomitant increase in severe erosion and social unrest) in thelate 20th Century.

The American Revolution. Historians recognize the paradox that theAmerican Revolution occurred among a largely prosperous people whoenjoyed a long period of economic growth. Furthermore, instead of beinga proletarian revolution of an oppressed underclass, the AmericanRevolution drew adherents from across the economic spectrum of colonialAmerica, including the poor, yeoman farmers, tenant farmers, merchants,intellectuals, and wealthy plantation owners.

By examining the distribution of wealth within the colonies, one cangain insight into the appeal of the revolution to people of greatlydifferent wealth levels. The economic history of the American coloniesup to the eve of the American Revolution confirms that, despite greatwealth differentials among colonists, they had as a group experienced along period of economic expansion and prosperity. This prosperity wasthreatened for all colonists by the imposition of various tax acts, andthe Quebec Act that frustrated colonial attempts to make legal landclaims west of the Appalachian and Allegany Mountains.

A representative sample of 919 individuals who lived throughout thecolonies in 1774 details their economic assets and liabilities.Examining the distribution of physical wealth (houses, equipment, realestate, slaves), it is clear that colonial America had an entirelyconvex distribution of wealth (FIG. 14). There were no clear wealthbased classes in the colonies, Furthermore, the wholly convexdistribution of wealth, in combination with the possibility of continuedeconomic growth through economic and territorial expansion, implies thatthroughout all wealth levels, people would generally have been riskprone over gains, standing to gain much by supporting the revolution.Lists of participants on Philadelphia revolutionary committees confirmthat the revolution drew substantial support from all wealth levels inthe colonies. Therefore, the broad appeal of the revolution can beexplained by considering the nature of colonial wealth distribution.

Contemporary American voting behavior. While driving to the polls isless dramatic than toppling a god-king or freezing at Valley Forge,researchers have established that the expected benefits of voting areless than the cost of voting, implying that voting is a risk pronebehavior. Who is investing their time and effort to vote, and who isdeciding that it is not to their benefit? Sociologists have longdemonstrated that the poor are less likely to vote, and that thispattern has been historically very stable. Sociologists also stress thatincome is the key variable in understanding voting behavior andpolitical participation. Given that Americans enjoyed a period ofunprecedented economic expansion prior to the 2000 election suggeststhat the sigmoid model evaluated over gains can provide insight into thepatterns of American voting behavior. The National Election Studies of2000 provide extensive and representative data on 1807 Americans' incomeand voting behavior. These data provide the additional opportunity ofexamining whether or not the actual individuals who fall on a convexwealth distribution curve were indeed willing to invest in voting.

Examining the wealth distribution curve, it begins concave and has aclear inflection between $25,000 and 30,000. The convexity becomesabrupt after $150,000 (FIGS. 15A-15B). Strikingly similar Americanwealth distributions can be observed in census data and in studies bythe Centers for Disease Controls (The initial concavity of the US wealthdistribution curve implies two things. First, social mobility from themost destitute to lower income brackets is abrupt. Second, the initialconcavity implies that the US economy is fueled by wealth drawn fromoutside the US national economy.). Therefore, one would expect voterturnout to increase with income, and to be higher than expected as theincome distribution curve becomes more convex. In fact, sharp increasesin voting behavior occur as income rises. Respondents with a familyincome in the lowest bracket ($0-2500) voted at a rate of 68% in thesample. Respondents whose family income was between $25,000 and 30,000were 86% likely to vote, and respondents whose family income was greaterthan $135,000 generally had 100% voter turnout.

A binary logistic regression to predict probability of voting based onincome level using SPSS 11.5 (available from SYSTAT Software Inc., ofPoint Richmond, Calif.) provided a highly statistically significant(P<0.00001) fit of expected and actual voting behavior (Log-of-the-oddsratio of voting behavior=0.934+1.89×10-5 (Income). R²=0.075. Allparameters and model are statistically significant at P<0.00001.).However, the fit is not very close (R2=0.075), and there are systematicdeviations from expected voting behavior consistent with theillustrative sigmoid model. Using the SPSS nonlinear regression routine,the following sigmoid function is fit to the wealth distribution data,including scaling of the data. The resulting equation isIncome=0.682×rank−sin(−0.707×rank), and its parameters are highlystatistically significant (P<0.00001), and close fitting (R2=0.914).This wealth distribution curve begins concave, inflects after$25,000-30,000, and becomes concave from $75,000 to 145,000. Aboveincome of $155,000, the data become strongly convex, requiring thefitting of another curve.

The sigmoid wealth distribution equation is used to calculate APmeasures of absolute risk aversion, and these measures correlated withthe residuals from the logistic regression. Risk averse decision makersshould vote less than expected (have negative residuals), and viceversa. The resulting data confirm this pattern, with a statisticallysignificant (P=0.007) and negative Pearson's correlation coefficient(r=−0.645) between residual voting behavior and AP risk aversionmeasures. In summary, voters with more to gain than to lose fromparticipating in elections voted more than their income alone wouldpredict. In contrast, voters for whom advancement would make less of adifference in their wealth and social standing did not turn out to voteas much as expected.

Ecological bounds exist on material wealth and these limitations implythat wealth distributions, and therefore, utility functions are bestmodeled with sigmoid curves. Individuals are not simply risk averse orrisk preferring, but their sensitivity toward risk is expected to varywith wealth in a predictable manner. Individuals within a step-likesocial stratum (middle class accountants, university professors, orblue-collar workers with secure union jobs) are expected to be riskaverse. Individuals between strata (starving animals, destitute poorpeople, upwardly mobile day traders, leaders of newly emergent worldpowers, adolescents) are expected to take chances, perhaps violent ones,because they have more wealth, status and prestige to gain if they win agamble than they will lose. Depending on the underlying probabilitydistribution of gambles, either AP or downside risk aversion measuresprovide reasonable predictions of behavior. Also, insights from prospecttheory, such as the influence of framing and loss aversion, need to beconsidered when modeling decision making.

In the cases of Late Classic Mayans, American colonists, and late 20thcentury United States citizens, upwardly mobile (i.e. considering gains)people whose wealth falls on a convex curve of a social material wealthdistribution are more likely to invest in or take a chance on increasingtheir material wealth and social status. Individuals whose wealth fallson a concave curve are less likely to make such investments and takesuch chances. These patterns have implications for both explanations ofpolitical conflict within a society, and for the design of developmentprograms. For instance, the Sicilian Mafia originated among higher-classpeasants, and neither among the landed elite nor the impoverishedpeasantry of 19th Century Sicily. Experimental agrarian developmentprograms appear to be more attractive to people who are neitherimpoverished, nor wealthy, but who are on the verge of entering a new(and higher) social class. Finally, revolutions are typically initiatedby disgruntled entrepreneurs and elites, rather than by the oppressedmasses. These cases indicate that further attention to step-like wealthdifferentials, ecological constraints, and their systematic influence onrisk sensitivity is warranted.

Modeling Effects of Global Wealth Distribution. In an earlier study ofwealth distribution in the U.S., a two-tiered economy appeared to exist.Wealth distribution among the wealthy was S-shaped and very steeplyincreasing. Wealth distribution among the rest of the population had ashallow slope and began concave. According to the inventors' theory ofsigmoid utility, the initial concavity of the U.S. wealth distributioncurve indicated that in this First World country, wealth was being drawnfrom elsewhere. Data presented herein demonstrates that the globaldistribution of wealth behaves as the inventors' theory of sigmoidutility predicts—it is initially convex, demonstrating the impoverishedand desperate condition of third world countries in relation to firstworld countries. Furthermore, there appears to be a two-tired globaleconomy, with an initially convex, shallow-sloped sigmoid distributionfor the third world, and a subsequent and initially convex, steep-slopedsigmoid distribution for industrial nations. The place of a nation onthese curves can lend insights into which nations (on the convex curves)have the most to gain by challenging the dominant world economic systemand which nations (on the concave sections) have little to gain bychallenging the world economic system. Importantly, the risk sensitivityof a nation's citizens and leaders is less influenced by their absolutewealth/poverty, and more influenced by their wealth level relative totheir nearest competitors.

In decision theory, an individual is expected to make choices thatmaximize his or her satisfaction, or utility. The standard neo-classicaleconomics assumption is that utility is wholly subjective. While utilityis subjective, its relationship to actual levels of wealth is thought tofollow certain functional forms, namely linear or concave (negativeexponential, power) functions. Utility is less a subjective individualphenomenon, and more influenced by social class and the distribution ofobjective wealth. Furthermore, wealth in a wide variety of economicsystems is always differentially distributed such that qualitative leapsbetween classes of individuals exist. Such discoveries imply anS-shaped, or sigmoid function that has alternating oscillations betweenconvex (concave upward) and concave (concave downward) sections.

The shapes of wealth distribution curves are informative since, asargued, utility is based on wealth distribution. Therefore, anindividual's satisfaction with wealth is based on that individual'swealth relative to others. If an individual is on a convex curve forwealth, then by definition, that individual has more utility to gain(the marginal rate of wealth acquisition is increasing) by taking a fairbet (50:50 of losing/gaining an equal amount) than what he or she standsto lose. Such a person is attracted to taking chances, or in other wordsis risk prone. Conversely, if an individual is on a concave section of awealth distribution curve, that individual would stand to lose moreutility than to gain in a fair bet. Such a person would seek insuranceagainst losses, or be risk averse.

The sigmoid model has been proposed to explain the influence ofsocioeconomic class on people's decision making and risk sensitivity.Subsequently, biologists and anthropologists have found the sigmoidmodel useful for explaining risk taking (risk proneness) and riskavoidance (risk aversion) in a variety of animal species and non-Westerneconomies. Empirically, not only are wealth differentials in human (andnon-human) societies ubiquitous, but they often contain qualitativeleaps in status or wealth that cannot be modeled by simple linear orexponential functions. These leaps can only be modeled by a sinusoidal,or sigmoid function. Furthermore, these complexities cannot be capturedby standard techniques such as Lorenz curves and Gini coefficients.

This theory has applications in global politics since nations, likeindividuals in a particular society, do not partake equally of globalwealth flows. Therefore, the risk sensitivity of citizens as well as theleaders of these nations, should be subject to the same influences fromwealth distribution that were identified among individuals in a varietyof economies. Nations that fall on a concave section of global wealthdistribution should be risk averse, and essentially play along withglobal economic influences since they have more to lose by not doing sothan they have to gain by rebelling. Conversely, nations on convexcurves should be risk prone, and willing to take risks (such as militaryaggression, overt challenges to the global economic market systemitself) in order to capture more wealth.

The key point is that poverty or wealth per se is not the determinant ofrisk sensitivity. Instead, it is the position on a sigmoid wealthdistribution that determines risk sensitivity. Therefore, it is entirelypossible to have an extremely impoverished nation where leaders andpeople nonetheless will not challenge their position, provided that theyare on a concave section of global wealth distribution. That is because,locally, they have more to lose than to gain from taking a risk.Conversely, it is possible to have a relatively well-off nation whosecitizens and leaders would be attracted to taking a risk because theystand to gain more wealth than to lose given their position on a convexsection of global wealth distribution.

Modeling Wealth Distribution. An illustrative sigmoid function has thegeneral form of: ax−sin(bx) where |b|≦a, and x is some measure of wealthor status. The linear parameter, a, measures the importance of good, x,in an economy. The sine parameter, b, measures the number of classes andthe degree of wealth differences among classes. The sine parameterallows the curve to oscillate around the line, ax, providing for thesigmoid shape. Since the sine component is periodic, the functionoscillates around quasi-periods of length b=2πk where k is an integer.

Estimating such curves to yield statistical significance tests andmeasures of goodness of fit is possible with non-linear regressiontechniques, for example, the routines in the SPSS family of software,available from SYSTAT Software Inc., of Point Richmond, Calif. Thesetechniques, based on maximum likelihood estimation, yield parameterestimates and standard errors that are asymptotically normal andtherefore can be used to calculate standard test statistics, providedthe number of cases is sufficiently large. Since the data set for theillustrative example contains 162 cases, the measures of statisticalsignificance should be robust.

Testing for Non-linearity. There are several methods for testing for thenon-linearity of data. Example methods that were used for theillustrative example were ANOVA and the Run's Test. In both cases, anOLS regression is performed on the data and the residuals are saved. Ifthese residuals systematically tend above or below the regression line,then the data are non-linear and a simple linear model, despite itstractability and fit, is inappropriate. If the data alternatelyoscillate above and below the regression line, then the data clearlyhave a sigmoid distribution. ANOVA is used by sub-dividing data into thesections that appear to systematically deviate above and below theregression line, and comparing the mean values of the residuals acrossthese classes. ANOVA has the advantage of being a parametric techniquethat utilizes interval data. The Run's Test is non-parametric and testswhether or not the sequence of runs (areas of sequential data above orbelow the regression line) is random or not. While this is a morequalitative test, it is more appropriate for the purposes since itmonitors oscillations above and below the regression line. In bothcases, ANOVA and the Run's Test yield highly statistically significantresults indicating that the data are both non-linear, and that thisnon-linearity oscillates in a sigmoid fashion. Therefore, a sigmoidmodel is preferred over earlier models, even if it may not yield asclose a fit as other models.

Measuring Population. The data on global wealth distribution are from a1990 compilation by the Economist (Samuelson 1990) and the 2001 U.N.Human Development Index study (U.N. 2001). These studies contain datafrom 1987 and 1999 respectively. Of the various measures off wealthprovided in these studies, one stands out as the best measure ofrelative wealth, the purchasing power parity measure (“PPP”). Thismeasure, comparable across nations, is the value of the per capitawealth (measured from gross domestic product (“GDP”)) in U.S. dollarsadjusted for the cost of living in a particular nation.

The Economist data cover 150 nations and the U.N. data 162.Unfortunately, data are aggregated by nation, hiding wealthdiscrepancies within nations. However, these aggregated data can stillprovide a sense of the global distribution of wealth over nations, and,with modification, over populations. Furthermore, since many poornations have only a few very wealthy families, the nationwide averagestatistics will provide a measure (admittedly imperfect and slightlyinflated) of the wealth of most of its average citizens. (NOTE: Thefigures were adjusted by using GINI coefficients provided in the UNdata. The adjustments had little effect, justifying the use of overallnation statistics).

Since the illustrative example concerns wealth distribution over people,not nations, the space each nation took up on the x-axis was adjusted byits relative population size as follows:

-   -   1. Rank the nations according to PPP.    -   2. Calculate the proportion of the world's population each        nation has.    -   3. Multiply the lowest ranking nation's rank by its population        proportion, then do the same with the second lowest ranking        nation, adding the previous value. Repeat this iteratively until        reaching the highest-ranking nation. This adjusts the various        nations' ranks to represent the populations contained by these        nations. This variable, hereafter referred to as population_rank        (xppp), is the x-variable that monitors world population.

This procedure provides a sense of how wealthy the average person is ina particular nation and how this wealth is distributed across the world.Therefore, the analysis provides a sense of how the people of a nationfeel about their relative wealth standing in the global economy. Anation's rulers, of course, will not necessarily share the view of theirpopulace. However, to the extent that political leaders respond to theneeds and aspirations of their people (and all must to some extent), theresults of this analysis may provide a sense of which nation states willbe more likely to challenge the world economy and which will be moreconservative.

Measuring Wealth. Due to the form of the sigmoid function, both therange and domain must often be scaled to allow the function's parametersto capture the variability in the data. The entire world data set wasscaled in the following way. The population rank (xppp) variable wasmultiplied by 10, and the PPP variable was divided by 750. This simplelinear transformation of the data in no way alters the relationshipbetween the variables.

Fitting Curves. As demonstrated below, while sinuous oscillationsclearly occur in the illustrative data, warranting a sigmoid model,there appear to be two qualitatively different relationships among thedata: there appears to be a section with a very gradual slope betweenpopulation and wealth, and another section with a steep slope. Thiswarrants the fitting of two separate curves to the data in theserespective sections (FIGS. 16 and 17). However, since all nations areintegrated in one world economy, the two curves should be joined andcontinuous. Following standard procedures in numerical estimation, thetwo equations should be equal at the node (the point where the twofunctions join), and furthermore their second derivatives should beequal at the node, and in this case zero. The illustrative method forfitting these two separate yet continuous functions is as follows:

-   -   1. The initial section appears to contain one complete        quasi-period. A period is defined by b=2π/x_(endpoint) where        x_(endpoint) is the end of the period for the data range.        Therefore, b is set equal to this value and the sigmoid equation        for the initial data range is estimated.    -   2. For continuity between the two sections of the data, both the        function values and their second derivatives should be equal at        the node. The point at which the two equations join is the node,        and this is the value (x_(endpoint), y) from the first equation.        The joint is formed by adding a y-intercept term, d, to the        second equation. Adjusting d allows the joining of the two        functions such that their values are equal at this point.

Since the illustrative sigmoid equation has a linear component and asine component, the linear component drops out in the second derivativeand only the sine component is left. Given that b=2π/x_(endpoint) in thefirst equation, the sine component equals zero at x_(endpoint), thenode. In order to guarantee this same value for the next equation, b isrequired to be an integer multiple of the first equation's b. In mostcases, this will yield an acceptable value of b for the next datasection, and in this particular case, it yields an extremely goodestimate of b.

Global Wealth Distribution in 1999. FIG. 16 illustrates the globaldistribution of wealth in 1999. This distribution is basically convex,but it contains sigmoid oscillations in certain sections. Wealth isconcentrated in a minority (about ⅓) of the world's nations. Thedistribution can be decomposed into two contrasting sections, and withineach of these sections a sigmoid relationship can be determined amongthe nations of that sub-section.

The Third World. The lowest ranking 106 nations (Sierra Leone toBrazil/Croatia) are placed on a decidedly shallow slope that foams theinitial and long tail of world wealth distribution. These nationsconform to what is generally called the Third World. The top ranking 54nations (Russia to U.S./Luxembourg) exist on a much steeper section ofthe wealth distribution curve and conform to what are generallyconsidered industrial and emergent nations. Examining these twosub-groups confirms that, while the world can be divided into haves andhave-nots in this way, the picture also has some interestingcomplexities.

After scaling the data for the 106 third World Nations, a very closefitting, sigmoid curve with one oscillation can be fit to the data. Thedata and the fitted curve are illustrated in FIG. 17. The best fittingequation is Adjusted PPP=0.927*xppp 10−sin(0.793*xppp 10). Theequation's R² is 0.784 and the model and its parameters arestatistically significant at well below the 0.00001 level. The curvebegins convex, as predicted for the world's most desperate nations(mostly located in sub-Saharan Africa and southern Asia). The curvebecomes concave in the middle, with nations such as India and China(these appear as the two “gaps” between data points).

While the estimated curve follows the general shape of the data, thereare a couple of informative discrepancies. First, the curveunderestimates the wealth distribution of the lowest nations, althoughit parallels their convex curvature closely, capturing the essentialrelationship. Second, the curve both under and over estimates the wealthdistribution for the very last section of the curve, which appears tohave a steeper slope than the previous nations. Further accuracy inprediction is possible by further sub-dividing the data set. However,for preliminary analysis, and for a general, global view of wealthdistribution, only two equations are estimated and joined for thisexample.

Interestingly, this last, steeply sloped section of the curve containssome of the world's most unsettled nations, whether measured by internalunrest or revolutionary politics. In Central and South America thisincludes Guatemala, El Salvador, Colombia, Peru, Venezuela and Brazil.In SE Asia this includes the Philippines, Fiji, and Thailand. In theArab world this includes Jordan, Syria, Lebanon, Kazakhstan, Iran,Tunisia, Algeria, and Turkey. In Africa, this includes Swaziland,Namibia, Gabon, and Botswana. In Eastern Europe, this includes Romania,Bulgaria, Belarus, Latvia, and Macedonia (there were no data forYugoslavia or Albania although one would predict their location here).With few exceptions, the nations listed above have arguably unsettledand frustrated populations, being on a steeply increasing curve (andconvex in relation to wealthier countries) on the border between thepoorest Third-world nations and the industrial world.

The Industrial World. FIG. 18 illustrates the wealth distribution andfitted curve for the world's wealthiest 54 nations. The best fittingcurve is: Adjusted PPP=15.654*xppp 10−sin(3.966*xppp 10)−116.675. Thiscurve has an extremely close fit with an R² of 0.923 and the model andall of its parameters are highly statistically significant well belowthe 0.00001 level. The curve begins convex, completes a period and endsconvex.

By far, the wealthiest nation is Luxembourg. Only five nations comprisethe wealthiest nations on the convex curve below Luxembourg. They arethe U.S., Norway, Iceland, Switzerland, and Canada. Below these nations,one sees a class emerge on a convex curve that goes from Ireland (thewealthiest of this middle class of nations) down through all of NWEurope and most of the other European countries. This section alsoincludes Arab countries such as Brunei, Kuwait, Saudi Arabia, Oman,Bahrain, Qatar, and developed Pacific countries such as Japan,Singapore, and Hong Kong. Interestingly, these represent Arab andPacific nations closely allied to Western, Industrialized nations in thecurrent conflict between Western nations and Islamic terrorists. Theinitially convex curve represents what many would call emergent nations.These include nations such as Russia, Mexico, Argentina, Malaysia, andLibya.

Referring still to FIG. 18, while a sigmoid curve provides a very closefit and agrees with the non-linearity in the data, the large value of arelative to b effectively linearizes the slope of the wealthdistribution curve. This nearly linear slope means that, whilevariations in risk sensitivity and class difference exist, there are nolarge, qualitative leaps in wealth from nation to nation among theindustrialized nations. Such a situation implies the possibility forsocial mobility by participating in the world economic system.Therefore, little social unrest within these nations or among them canbe predicted, as is the situation in nearly all cases. While thesenations certainly complete hotly with one another for wealth, they tendto do so in the arenas of diplomacy and economic competition, ratherthan revolution and military aggression. Overall, the fitted curvespredicting world wealth distribution in 1999 are illustrated in FIG. 19.

The Economist provides data on PPP distribution for 1987, just beforethe end of the Cold War. The overall distribution is much like that in1999. However, some informative differences exist when one comparesindustrial nations for the pre- and post-Cold War periods. FIG. 20illustrates the distribution of wealth among industrialized nations in1987 and the best fitting curve. The best fitting curve is: AdjustedPPP=4.18*population−sin(2.14*population).

The data comprise the wealthiest 36 nations in 1987. The curve has an R²of 0.965, statistically significant at well below the 0.00001 level. Thelinear parameter is likewise statistically significant at well below the0.00001 level, and the sine parameter is statistically significant atthe 0.003 level. Interestingly, before the fall of the Soviet Union, thesigmoid curve begins convex (as predicted) and completes two periods,ending concave. Such a distribution indicates that the supply of wealthis nearing exhaustion. A similar argument based on historical trends inagricultural growth has been advanced by others. The Cold War may havebeen draining the pace of economic growth by, this point in history.

The curve begins convex, as expected, and with many of the same nationsseen in the 1999 data, including Mexico as an emergent nation. However,interesting changes have occurred in the positions of the top rankingnations. In 1987, the U.S. was ranked at the top, followed by Canada,Switzerland, Norway, Luxembourg and then Iceland. The first strikingdifference in the 1999 data is that the curve now ends convex, and theoverall curve is more linear, only having 1.5 oscillations. Thisindicates that more wealth has been created since the Cold War, thatopportunities to share in that wealth among Industrial nations haveincreased, and that 1999 was a period of genuine wealth expansion.Second, the U.S. and Canada have dropped in rank since the Cold Warwhereas the Scandinavian countries have increased their wealth ranking.Third, Luxembourg not only makes the most dramatic increase in wealth,but in 1999 it had a PPP ($42769) that was extremely higher than anynation, even the second-ranked U.S. ($31872). This clearly begs thequestion of how Luxembourg has managed to benefit in the world economysince the fall of the Soviet Union.

In general, the fall of the Soviet Union would have loosened upinvestment opportunities in the former Soviet-block countries. China hasalso seen an expansion of foreign investment since the end of the ColdWar. So, in part, the increased generation of wealth can be attributedto capitalist expansion into these populations. Profit-orientedenterprise appears to generate more wealth through increased efficiencyof production and exchange. However, considering the low rankings ofRussia and China in 1999, there appears to be a net flow of new wealthtoward Industrial nations. Should this situation continue to thedetriment of the people and leaders of these nations, wealthdistributions would shift them into more convex positions, making themmore likely to take greater chances (as in aggression) to claim theirportion of global wealth.

Of course, the increase in wealth in 1999 may also be a reflection ofthe dot-com and other corporate bubbles, and of increased energeticefficiencies during the 1990's.

Another point of interest, the wealthiest Arab nation in 1987 wasKuwait, and it was the 7th wealthiest nation, per capita, in the world.Perhaps it is not a coincidence that the U.S. took such interest in thisally's fate when it was attacked by Iraq.

An Expo-Sigmoid Model. As discussed above, while basically sigmoidwealth distributions are common, complex societies in which elitescontrol a majority of the wealth exhibit an overall exponentialdistribution of wealth with sigmoid fluctuations around this trend (FIG.17). The following extension of the above formulation models suchdistributions with a single functional form, avoiding singularitieswhere two separate curves join. The resulting illustrative expo-sigmoidwealth distribution equation is W(x)=e^(d+ax+c sin(bx)), a>bc. W is usedfor this function to avoid confusion with traditional utility functions,although a systematic relationship between W and the U of utilityfunctions holds. The parameter, a, is the same as in the sigmoidfunction above. Parameter c is an additional parameter that allows foramplitude in the sigmoid fluctuation around the underlying exponentialdistribution. The constant, d, is the natural log of the y-intercept ofthe expo-sigmoid distribution. Applying this model to the world wealthdistribution provides a close fit and yields the following equation:W(x)=e^((−0.438+0.4x+0.399 sin(x))). All coefficients are significant atthe p<0.0001 level, R=0.966.

An Expanded Expo-Sigmoid Methodology. Using standard non-linearregression techniques to fit curves to the expo-sigmoid formulation isdifficult since the routines are very sensitive to parameterinitializations, and often do not converge. The following method forfitting the expo-sigmoid model to actual data sets effectively sets b=1,providing a simple periodicity of 2π:

-   -   1. Take the natural log of wealth data to remove exponential        effects.    -   2. Perform an OLS regression to extract the linear effects from        the data, resulting in an equation d+ax, and residuals. Given        the sigmoidal fluctuation around the exponential. curve, these        residuals exhibit a degree of periodicity.    -   3. Most data samples are not uniformly spaced over the        population. Lomb's periodogram method provides a means of        determining the harmonicity influence at each data point in such        a non-uniform sample while being able to eliminate independent        Gaussian white noise in the data. Apply the periodogram        algorithm to this data to determine the trigonometric polynomial        that provides a least squares fit to the residual data. The        resulting best-fitting equation to the data is of the form:        d+ax+Σ_(j) A_(j) cos(f_(j)x)+B_(j) sin(f_(j)x), where each f_(j)        is a statistically significant frequency influencing the        periodicity with amplitude A_(j) and B_(j) that results in the        sigmoid variation in the residuals.    -   4. Construct an argument for the expo-sigmoid function by using        the best fitting trigonometric polynomial from the periodogram        analysis as an argument to the exponential function:        W(x)=e^(d+ax+Σ cos(x)+sin(x)).

Simulating the Effect of Nepotism on Political Risk Taking and SocialUnrest. Nepotism has been the primary influence on political behaviorthroughout human history. Despite the spread of democracy in the 20thcentury, nepotistic regimes have hardly disappeared. Nepotism heavilyinfluences political activity throughout the developing world, MiddleEast, and central Asia where family ties are essentially for gainingaccess to power, state resources, and privileges. Rebelling against suchnepotistic regimes is difficult and risky. The above disclosedillustrative ABM system can be used for testing the influences ofvarious social forces on risk taking behavior, including the formulationof rebellious coalitions. The illustrative ABM system is also used toexamine the influence of nepotism on the distribution of wealth andsocial status. Nepotism heavily skews the distribution of wealth andstatus, leading to the formation of opposing coalitions and exacerbatingsocial unrest.

Political activity is influenced by many factors, including the pursuitof wealth, quest for power, and assertion of political and ethnicidentity. Modern Western democracies admit of all these influences, buttend to de-emphasize the role kinship plays, and should play, inpolitics. However, anthropologists who work in non-Western societiesrecognize the central role kinship plays, and has historically played,in human political activity. Given their western, democratic bias,Westerners tend to ignore the important role kinship continues to playin the politics of many non-Western states.

Nepotism dynamically shifts the distribution of wealth and social statusin a social system, increasing the potential for risk taking and socialunrest. The analysis of nepotism not only has relevance to tribalpolitics and the rise and fall of ancient Kingdoms, but also sheds lighton aspects of social unrest in patriarchically dominated modern statessuch as Saudi Arabia, and Uzbekistan.

Nepotism is the favoring of kin, in comparison to others. In many ways,nepotism is the first principle of human political interaction. It isonly in developed democracies where nepotism laws are enacted,prohibiting the preferential assistance of kin. In the simplest humansocieties, hunter-gatherer bands, sibling alliances stand at the core ofband political organization. In tribal Amazonian societies, kinshipalliances structure intratribal violence and intertribal warfare. Inmore complex chiefdoms, powerful lineages not only come to dominatetheir own groups, but eventually dominate others. In the context ofmodern conflicts, nepotism is often a contributing cause of grievanceand conflict. Anthropologists have described the connection betweennepotism, xenophobic rhetoric, and modern political mobilization.Nepotism, alliance building and conflict extend to the analysis ofstreet gangs and lynchings. Nepotism has figured heavily in popularviolence and policing in Northern Ireland. Nomadic pastoraliststypically distribute aid along kin lines, as among the Bedouin Arabs.Nepotistic favoritism among powerful Arab merchants and politicalleaders contributed to the current conflict and humanitarian crisis inDarfur, Sudan. Finally, modern states such as Saudi Arabia (ruled by thehouse of Saud) and Uzbekistan are wholly dominated by family ties.Nepotism is a pervasive tendency in human behavior, and any attempt atunderstanding human interaction must take it into account.

The wealth variable in the illustrative ABM system can be considered anyculturally appropriate form of wealth or power over which individualswould compete in a particular society. Anthropologists have clearlydemonstrated that wealth equality has never existed, even in thesmallest and simplest hunter-gatherer societies. Status-conferringwealth in the simplest societies, such as hunter-gatherer bands, istypically distributed in a simple sigmoid fashion, which is best modeledas a linear increase with sigmoid fluctuation around this line, and asimple sigmoid wealth distribution can be used to initialize theillustrative ABM system for nepotism as illustrated in FIG. 22. Agentsin convex (concave upward) sections have more to gain than to lose whentaking chances, and are therefore risk prone. The symmetry of thedistribution indicates that there is parity between risk prone andaverse agents. Using standard measures of risk sensitivity, such as theAP measure, a society-wide risk sensitivity that is zero, or riskneutral is achieved.

However, in complex societies, wealth distributions are basicallyexponential, with sigmoid oscillations around the exponential curve, asillustrated in FIG. 3C. Interestingly, the sigmoid fluctuations dominatethe risk sensitivity of agents, producing alternating risk prone andaverse agents despite the overall concave convexity of the curve.However, the overall convexity of the curve increases the risk pronenessof the population, yielding an overall negative measure of risksensitivity. The key question is what dynamic creates the expo-sigmoiddistribution of wealth, and how is it likely to affect the risksensitivity of agents, and their coalition forming behavior. Agentcoalition formation and its payoffs is modeled with the coordinationgame, which is described at length herein above for other analyses.

If nepotism is instantiated, then agents must have kin. For theillustrative ABM system, agents are allowed to reproduce at a rate of 2%per annum. Nepotism occurs in two ways. Children benefit from parents byhaving a higher probability of being born, and by inheriting aproportion of their wealth. Fertility is a function of wealth andstatus. Fertility is increased with wealth, and offspring inherit 50% ofa parent's wealth during any iteration. Fertility is increased byincreasing an agent's probability, of reproducing (Pr.Reproduce)=wealth(i)/MAX, where MAX=highest status agent's wealth. Also,an agent's payoff is a product of fitness, such that:Payoff=payoff*0.25(kin), where kin=inclusive reproductive fitness. Inthis manner, both the benefits of being born into a large, wealthylineage, and the recursive benefits to lineage members of the additionof another agent to their kindred are captured.

The effects of nepotism transform a simple sigmoid wealth distributioninto the classic expo-sigmoid distribution seen in complex societies.The resulting distribution has distinct concave and convex sections FIG.23. These sections are created in two ways. First, the abundantdescendants of high-status individuals tend to accumulate as asecond-tier “noble class,” reproducing the demographic seen in Kingdoms,and ancient states. Second, relatively impoverished agents tend toremain poor, as well as their offspring, leading to the long “tail” inthe distribution of wealth.

The few convex sections of the curve are those areas where one wouldexpect a high level of displeasure with the status quo, and consequentlythe source of social unrest. Furthermore, the overall convexity of thecurve yields an overall risk prone, negative risk sensitivity value. Themost risk sensitive agents are those between 0 and 250, and those around1200. A distribution such as this produces alternating and strongly riskaverse and risk prone agents. Research confirms that agents in convexsections tend to aggregate in coalitions, and these coalitions arepotentially volatile due to the risk proneness of their agents.Empirical research confirms that agents in such real world coalitionsare responsible for social unrest and collective violence, includingpolitical coups, and more popular rebellions.

Recent events in Uzbekistan provide an illustration of just how thesedynamics can lead to political upheaval. Nepotism and abuse in theregime of Islam Karimov sparked an armed demonstration. For now, theUzbek government has been able to stay in power; however the conditionsthat led to this social unrest will strengthen due to the recursiveeffects of nepotism in skewing wealth distributions.

Illustrative input data sets and ABM simulation system output. FIGS.24-32 represent illustrative input and output data for the illustrativeABM system to simulate agent interaction in Palestinian society.Specifically, FIGS. 24A and 24B are a data table of Palestinian wealth,and FIG. 25 is a scaled and sorted graph of the data drawn from thetable and used as input data for the initial distribution of resourcesfor the simulation. FIGS. 26-32 illustrate aspects of deriving anexpo-sigmoid function for the scaled and sorted input data.

FIGS. 33-37 and 38A-38F illustrate an input data set and simulation dataoutput for the illustrative ABM system. Specifically, FIG. 33illustrates an initial wealth distribution according to a derivedsigmoid distribution W(x)=15x−400 sin(x). FIG. 34 illustrates thecalculated join probability by wealth. FIG. 35 illustrates the APmeasure of risk sensitivity by wealth rank. FIG. 36 illustrates thewealth mobility (the difference between an agent's lowest rank andhighest rank in a simulation) of agents according to wealth rank. FIG.37 illustrates the volatility, or the sum of the absolute value ofwealth gains and losses for a simulation.

FIGS. 38A-38F illustrate the rank history for specific iterations ofgame play for various agents. FIG. 38A illustrates agents beginning atranks 1 and 16. FIG. 38B illustrates agents beginning at ranks 17 and22. FIG. 38C illustrates agents beginning at ranks 112 and 128. FIG. 38Dillustrates agents beginning at ranks 256 and 272. FIG. 38E illustratesagents beginning at ranks 288 and 304. FIG. 38F illustrates agentsbeginning at ranks 496 and 512. These ranks were chosen to illustraterepresentative differences in agent behavior across the simulatedpopulation.

The illustrative ABM system as a Theoretical Research System. Rationalchoice theorists stress that norms and group membership exist becausethey serve the material needs of the individuals within groups. Forinstance, the formation of territorial groups occur when resources arepredictable and dense, and therefore profitable for individuals tocooperate defending them. In contrast to the common Prisoner's dilemmagame, coordination games in which solutions include both mutualcooperation and mutual defection are proposed as better models of manyhuman interactions. They demonstrate that optimality analysis providesthe range of solutions decision makers can choose, although otherconsiderations may be necessary to explain which Nash equilibriumsolution people actually choose.

In contrast, bounded rationality theorists argue that optimizationrequires an unreasonable amount of information and cognitive ability,and instead people use fast and frugal heuristics to make decisions.Furthermore, they maintain that there is no global decision rule (likeoptimization), and that people use varied simple heuristics attuned tospecific environments. Altruistic heuristics could be selected for ifcooperation is necessary to produce a public good, allowing members ofgroups of cooperative individuals to benefit and reproduce more thanmembers of groups of selfish (and therefore less productive) members.They argue that altruistic heuristics could explain the evolution ofethnic cooperation, and account for ethnocentrism and xenophobia, andthat the evolution of arbitrary ethnic markers can enhance coordinationamong group members. Economists and anthropologists have used theconcept of cultural group selection to explain patterns of cooperationin experimental bargaining games, deference to authority, and theadoption of innovations.

Simultaneous testing of these decision rules is especially importantbecause of the dynamic nature of competition. A Nash equilibriumsolution (and a game may have more than one) is optimizing. However,Nash equilibrium is based on the assumption that your opponent is asrational and intelligent as you are. If an opponent is non-rational(considered a mutant strategy), then that assumption is no longer valid.An evolutionarily stable strategy (ESS) is one that cannot be beaten bysuch mutant invaders. All ESS's are Nash equilibria, but not all Nashequilibria are ESS's. Fast and frugal heuristics are non-rationalstrategies, and so a true test of these competing claims requires amechanism such as the illustrative ABM system for testing the long-runadaptability of different decision rules against each other. Thebackdrop of these theoretical debates provide four basic hypotheses,various aspects of which can be tested with the illustrative ABMsimulation system.

Hypothesis 1: Rational Choice: Agents using Nash equilibrium to solvecoordination problems will form coalitions that are well adapted(provide competitive material benefits to their members) under a varietyof environmental conditions.

The use of Nash equilibrium in non-zero-sum payoff games is a goodexample of rational choice optimization for coordinated games. In orderto achieve Nash equilibrium a player must be able to understandeverything needed to know to make an optimal decision, and alsoappreciate everything the opponent understands as well. Given theinteractive nature of sociability and coordination games, Nashequilibrium is selected as the decision rule to model rational choice.

Hypothesis 2: Boundedly Rational Conformist Transmission: Agents usingconformist transmission rules will form effective coalitions based on anorm of imitating the most common behavior of their neighbors. Thesecoalitions will be adaptive in stable environments.

One of the boundedly rational tools relevant to the formation of ethnicnorms is conformist bias, or conformist transmission. Conformisttransmission involves basing ones actions and beliefs on what is mostcommon in a population. Conformist transmission is argued to beadaptive, provided that environments are stable and do not change toorapidly. It is also argued that conformist transmission is involved inthe adoption of innovations and attitudes toward risk, and it plays arole in the evolution of ethnic markers.

Hypothesis 3: Boundedly Rational Prestige Bias. Agents using prestigebias will form coalitions based on the norm of doing what theirwealthiest neighbor does. These coalitions will be very dynamic, andchanging norms within a group are expected as individual fortunes waxand wane. Therefore, prestige bias may be particularly effective inrandomly fluctuating environments.

Prestige bias is another bounded rationality proposition related to theformation of coalitions and norms, and refers to the imitation of highstatus individuals. Researchers have proposed that prestige bias canexplain a wide range of behaviors such as fads, innovation adoption,deference to high status, and the evolution of ethnic markers.

Given the competing claims of existence and efficacy of these variousdecision rules, the illustrative ABM simulation system may be used fortesting the following hypothesis in order to see which rules are likelyto have survived and become part of the human behavioral repertoire.

Hypothesis 4: If information constraints exist, then boundedly rationaldecision heuristics would be expected to outperform optimization rules,with conformist bias dominating in a stable environment, and prestigebias dominating in a fluctuating environment. However, it has been notedthat in dynamic models involving competing strategies, the strategiesthat eventually win out tend to be Nash equilibria. The illustrative ABMsimulation system under information constraints provides a test of therobustness of Nash equilibrium solutions.

Further Hypotheses Considered: Space prohibits full exploration of allcombinations of proposals in the literature. For instance, the importantrole language plays in pre-game communication, allowing players toachieve optimal solutions. Also, coordination games come in a variety oftypes such as Battle of the Sexes, Chicken, and Assurance, and a fullanalysis should consider the effects of altering payoffs to model thesevaried scenarios. Furthermore, modeling the effects of learning in anyrealistic repeated game scenario. The illustrative ABM simulation systemallows modeling of all these variants, beyond the several basichypotheses outlined herein.

A recent example of modeling ethnic interaction with coordination gamesillustrates the promise and limitation of current attempts. One toolmodels the extent to which norms (shared behaviors) become marked(indicated by some symbol) based on agents playing coordination gameswith two Nash equilibria. Difference equations are used to measure thechanging frequencies of norms and markers in order to model thedevelopment of ethnic groups, defined as those whose members share bothnorms and markers. The model assumes that agents use prestige bias orconformism to decide upon which strategy to employ, and parametersmeasuring their effects are introduced into the model. The modelingbegins by pre-defining groups, and then monitoring how changes inparameters would influence the formation of groups with norms, groupswith shared markers, and groups with both. The mathematics of thedifference equations is intractable so a computer simulation is used tomonitor the effects of changes in their parameters.

The model yields the following results: differences in shared behaviorsusually become marked, increasing number of populations increaseslikelihood of markers, and group differences are greatest at boundaries.These findings are suggestive, but both the framing of the model and themethod of analysis have drawbacks that preclude genuine testing ofcurrent theories and limit the realism of the model.

First, some of the model's expectations are anticipated by theconstruction of their model. For instance, the model assumes thatindividuals tend to interact with others who have the same variant ofmarker trait, and offers the parameter, e, to measure this effect. Itseems that assuming the existence of such a propensity and introducingit into the model guarantees the basic expectation that norms will form.Also, groups are predefined, which sets the stage for inter-groupdynamics, and drives the modeling results in the expected direction.Finally, the model incorporates only the rules thought to exist, and notalternative boundedly rational rules or optimization rules, precluding atrue test of which rules are most adaptive. The prefiguring of results,along with the lack of alternative decision rules in the model, weakensthe ability of the model to provide a genuine test of competing claimsregarding how and why ethnic groups evolve.

Another limitation is that the model characterizes whole populations bypropensities that are assumed equal for all individuals within apopulation. For instance, the propensity to interact with others of samemarker, migration rates, and rates of random borrowing of norms andmarkers, are all assumed equal across populations for any simulation.While doing so gives a general sense of how average individuals arelikely to interact, in reality populations are made up of individuals,not averages. Analyzing behavior through aggregate means may elidelocalized phenomena and short-term variances as well as chaotic-likeperformances or bifurcations in system response.

Modeling individual differences requires discontinuous models, which areextremely difficult to analyze mathematically, thus the immediatelyaboved discussed research model has limitations. Overcoming theselimitations is daunting, since a more appropriate methodology wouldinvolve simulation of discrete interactions among individuals in arealistically large population with the use of powerful parallelprocessing computers with tremendous storage space (measured intetrabytes).

The illustrative ABM simulation system provides a more realisticsimulation that will allow the modeling of discrete interactions among alarge population of individuals under a variety of environmental andsocial conditions, and the testing of competing theories of humandecision making and ethnic group formation. One illustrative ABMsimulation system provides a small-scale simulation on a desktopcomputer, and provides evidence for coalition generation with theillustrative simulation algorithm (see below), providing proof ofconcept and demonstration of the potential for the larger simulation ona high-end computing cluster. Such a cluster, each with large amounts ofrandom access memory and secondary storage fully networked viafiber-optic, provides computing power to test several competing theoriesof decision making by demonstrating how well they can account forcoalition and norm formation. In addition, differences in howenvironmental structure and information constraints impact theperformance of each kind of decision rule can be examined. Running theillustrative simulation algorithm on such a cluster also supportslarge-scale simulations in which these rules are pitted against oneanother in order to see which is most competitive or survivable throughtime, and therefore more likely to account for the evolution ofethnicity in human groups.

The illustrative simulation system, also described in part above andbelow, includes a large number of agents, akin to cellular automata,distributed in an environment and having access to a resource necessaryfor survival. Each agent has to play a coordination game with itsneighbors via a decision rule (Nash equilibrium, conformism, orprestige-bias rule, depending on the simulation) results in eithercooperation and therefore coalition formation, or non-cooperation andtherefore coalition defection. Such decisions should lead to theformation of coalitions if agents are able to garner more resourcesthrough cooperation than defection. All cells, constituted as a largegrid on a two-dimensional surface, are mapped onto the surface of asphere to avoid the influence of boundary conditions. Finally, the cellsadjacent to an individual agent/cell constitute the neighborhood forthat agent/cell. Resources are relevant to the game play, and constitutea hazard for players since agents may be eliminated from the simulationwhen their individual resources are depleted.

Simulation System Methodology. The following symmetric, nonzero-sum gameis a modification of the Assurance Game central to models of strategycoordination and group formation. In this model, R is the reward forcooperation, P is the punishment for not cooperating, T is thetemptation to defect, and S is the sucker's payoff if one's partnerdefects while one cooperates. In the Assurance Game, R>P>T>S. Thesegames are referred to as Stag hunt games based on an analogy proposed byJean Jacques Rousseau. In a stag hunt, hunters must coordinate to kill astag and reap a large payoff, but hunters can hunt hares alone for asmaller payoff. Considering the analogy further, if one agreed to huntstags but one's partner defected and hunted hares alone, the suckerwould receive nothing, but the defector should still receive the samepayoff as if both went their separate ways hunting hares. In otherwords, P=T. Therefore, in the illustrative simulation algorithm, thepayoffs are R>P=T>S (FIG. 5) to model typical human coordinatedinteraction situations.

The optimal, mixed strategy for both Row and Column players is (⅗ Join,⅖ Defect). A Join coalition is counted as all those agents that playedJoin during the current game, regardless of their payoff. The optimalmixed-strategy (Nash equilibrium) produces a greater amount of resourcefor two players who play Join-Join. This biased payoff should give along-term advantage to coalition formation, when players are playing theoptimal mixed strategy.

Following a coordination game framework, the Defecting players receive areward, albeit less than Joining players, and there is a greaterindirect penalty for the Joining player who plays a Defector. The Joinerwill gain nothing in the play with a defector. A resource depletionthrough a metabolic tax payment during each round of game play andsuccessive zero payoffs could cause some players to “die” from resourcedepletion. Agent-players must have non-zero resources to play. Animportant point to note is that previous analyses of ethnicity assumethe existence of groups. In the illustrative simulation algorithm,groups that share behaviors and develop norms evolve on their own,providing a further test of the efficacy of current theories; if adecision rule never allows formation of ethnic groups, then it probablyis not valid.

The illustrative ABM algorithm 100 shown in FIGS. 7A-7D implements theabove discussed algorithm and can be used for evaluating theoreticalmodels by using variations from the basic decision rule and analyzingresults, for example as described below.

Variation in Resources Experiment. Resources are distributed unevenly,creating pockets of wealth and poverty. Resources are randomlyfluctuated through time, creating unstable environments.

Conformist Strategy Experiment. A conformist agent strategy is selected.Initialize agents by flipping a fair coin to choose first strategy.Thereafter, agents chooses most-used strategy among neighbors duringlast iteration. If there is a tie, break arbitrarily by a fair coinflip.

Prestige Bias Experiment. The agents are initialized by flipping a faircoin to choose first strategy. Thereafter, agents choose dominantstrategy of neighbor with most resources at end of last iteration. Ifthere is a tie, break arbitrarily by a fair coin flip.

Sensitivity Analysis. Modify the payoff matrix in the coordination gamewith small increases/decreases in Join reward and defect rewardsseparately. Rerun the above variations and compare new results forsimulation dependency or over-sensitivity to payoff entries in gamematrix. Note: In the basic and first variation of the simulation,re-calculate Nash equilibrium mixed strategy. Use these results to plotresults from changes in matrix entries, and use regression curve fittingtechniques to analyze any trends in system response.

Information Constraint Experiment. Perturbed Matrix for InformationConstraint Experiment. Information constraints are modeled by randomlyaltering the payoff matrices agents use to formulate decisions, butpaying out the standard payoffs. This models an agent's imperfectinformation about the payoff-generating environment. For example, alteran agent's payoff matrix by replacing the cooperation payoff, R=5, witha random variable, a≧3, which alters the probability of cooperating from3/a to near zero, therefore altering an agent's choice. Agent flipsbiased coin according to these new probabilities, but is rewardedaccording to the actual environment of payoffs in the originalcoordination game.

Method of Analysis. Fourier analysis of the time series generated by thenumber of cells in a coalition is used. Fractal behavior of coalitionsmay be detected in this iterated function system, such as thecellular-automata-like structure, through a plot of the Fourier analysispower spectrum.

Model Expectations. The basic model expectations for hypotheses 1-3 areas follows: Hypothesis 1. Rational Choice. The coordination strategyshould dominate through time and lead to the formation of coalitions.Given the theoretical long-term stability of Nash equilibria, coalitionsshould form under a wide variety of environmental conditions (unevennessof resources, instability of resources), and potential payoffs.Following studies of cellular automata, dynamic behavior in coalitionsize in both frequency and amplitude is expected. However, becauseagents are randomizing between Nash equilibria, a somewhat cyclicalresponse that will be evident in Fourier analysis is expected. Since acoalition's boundaries influences its next size, fractal analysis isused to monitor time-dependent, distributions of increments in coalitionsize to gain further insight into fractal patterns in coalition sizefluctuations and the frequencies at which they may fluctuate.

Hypothesis 2. Boundedly Rational Conformist Transmission. Conformisttransmission is expected to be most adaptive in stable environments, andto lead to a breakdown in coalition formation in unstable environments.There is no reason to expect conformism to break down in ageographically uneven environment because agents are imitating mostcommon adaptive behaviors in response to local resource levels.Conformist transmission should have an averaging effect on coalitionsize fluctuations because agents are imitating the majority of theirneighbors. Therefore, lower amplitudes are expected in the Fourieranalysis of coalition size, and fractional Brownian motion measuressmaller in relation to other decision rules.

Hypothesis 3. Boundedly Rational Prestige Bias. Coalitions are expectedto form based on prestige bias. Given fluctuations in individual'swealth the opposite of conformist transmission's averaging effect isexpected. The behavior of coalitions in this system are expected toexhibit greater amplitude as measured through Fourier analysis, and ahigher fractal response than other rules.

Referring to FIG. 21, basic model expectations are illustrated by cellentries that refer to strengths of effects relative to differentdecision rules.

Hypothesis 4. Information Constraints. Testing the Adaptiveness ofAlternative Decision Rules and Rational Choice vs. Bounded RationalityPrograms. Limiting environmental information should decrease theadaptiveness of optimizing techniques relative to a perfect informationenvironment, and relative to boundedly rational decision rules. However,how much one must interfere with the environment to achieve this affectwill gauge how profound ignorance should be in order to select forboundedly rational decision rules. Should information constraintsrequire near or total ignorance to make boundedly rational rules moreadaptive, then this would cast doubt upon the bounded rationalityprogram. In contrast, if little perturbation is required, then therational choice program would be questioned. The methods above can beused to explore the further effects of varying environmental structureon the performance of the rules, and gauge cycles and fractal effects.

There are many practical applications for research into coalitionformation. For instance, the post-Cold War era has seen the unexpectedrise of ethnic identity and strife on a global scale, even whenmaintaining ethnic distinctions appears counterproductive to those whomaintain them. This proliferation of ethnicity poses a historicalparadox. Anthropologists have proposed theories to explain, inhindsight, ethnic formation as a response to economic and politicaldomination. However, other periods of history are marked by the collapseof ethnicity and cultural homogenization in the face of economic andpolitical domination, considering examples as diverse as the ancientSumerians, the Inca Empire of South America, and westward expansion inU.S. history. How can one explain the rise of ethnic coalitions todayand the dissolution of ethnic coalitions in the past? To date there areno singular social theories that effectively explain both the formationand dissolution of coalitions. Research using the illustrative ABMsimulation system will allow a better appreciation of both phenomena.Political machinations within nation states involve the same coalitionformation, but on a smaller scale. Likewise, clique formation amongteens (all-important in their lives) and alliance formation insmall-scale human and non-human primate societies involve the samecomplex combination of strategies, levels of understanding, and chance.Therefore, the illustrative ABM simulation system described herein,based on the best understanding of the cognitive tools humans have attheir disposal, can inform a very wide range of issues in the social andbehavioral sciences.

Although certain illustrative embodiments have been described in detailabove, variations and modifications exist within the scope and spirit ofthis disclosure as described and as defined in the claims.

1. A method in a parallel processing computer system for simulatingindividual and social behavior, comprising: creating a neighborhood ofcells; allocating resources to the cells; assigning an agent to eachcell; selecting one of a plurality of decision rules for interactionsbetween agents, wherein one of the plurality of decision rules includesa coordination game; analyzing risk sensitivity using an analyticalfunction W(x)=e^((d+ax+c sin(bx))), where a>bc; iteratively conductingdiscrete interactions between the agents using the selected decisionrule and allocated resources, wherein the discrete interactions of afirst subset of the agents are conducted on a first computing device ofthe parallel processing computer system and the discrete interactions ofa second subset of the agents are conducted on a second computing deviceof the parallel processing computer system; and recording interactionoutcome.
 2. The method of claim 1, wherein the coordination gameincludes a mixed strategy matrix of: Join Defect Join R, R S, T DefectT, S P, P

wherein payoffs include R>P=T>S.
 3. The method of claim 2, wherein R=5,P=3, T=3, and S=0.
 4. The method of claim 2, wherein R, P, T, and S aremodified with small increases and decreases for payoff sensitivityanalysis.
 5. The method of claim 2, wherein P=3, T=3, S=0, and R is arandom variable ≧3.
 6. The method of claim 1, wherein one of theplurality of decision rules includes a conformist strategy.
 7. Themethod of claim 1, wherein one of the plurality of decision rulesincludes a prestige-bias strategy.
 8. The method of claim 1, wherein oneof the plurality of decision rules includes a modified risk posturebased on at least one of a sigmoid and an expo-sigmoid function fordistribution of resources among agents.
 9. The method of claim 1,wherein allocating resources includes uniformly allocating resources tothe cells.
 10. The method of claim 1, wherein allocating resourcesincludes allocating resources according to a function of a data set, thefunction including at least one of a sigmoid and an expo-sigmoidalfunction derived from Fourier signature analysis fit of residuals fromat least one of a linear regression to the data set and a logarithm ofthe data set.
 11. The method of claim 1, wherein the one of theplurality of decision rules includes a join probability according to afunction of a data set, the function including at least one of a sigmoidand an expo-sigmoidal function derived from Fourier signature analysisfit of residuals from at least one of a linear regression to the dataset and a logarithm of the data set.
 12. The method of claim 1, whereinat least one of the resources and the one of a plurality of decisionrules includes a function of a data set, the data set including ameasure of at least one of well-being, social status, and social worth.13. The method of claim 12, wherein the measure of at least one ofwell-being, social status, and social worth includes a measure of atleast one of demographic data, income, net worth, wealth, personalproperty ownership, real estate ownership, social prestige, mortality,morbidity, marital status, family status, sex, age, education, health,literacy, human development index, geography, environmental factors,movement, communication, informational constraints, mass media,infrastructure, social services, materials, conflict, peacekeepingforces, provocateurs, influential agents, risk sensitivity, culturalnorms, ethnic affiliation, ideological affiliation and kinship.
 14. Themethod of claim 12, further comprising analyzing recorded interactionoutcome.
 15. The method of claim 14, wherein analyzing recordedinteraction outcome includes determining an effect on risk sensitivityof the measure of at least one of well-being, social status, and socialworth.
 16. The method of claim 14, wherein analyzing recordedinteraction outcome includes determining at least one of an individual'slikelihood of taking a risk, a distribution of well-being, adistribution of social status, a distribution of social worth, apotential for social unrest due to wealth inequalities, an impact ofnepotism on social unrest, an influence of class origin of insurgencymembers, and an influence of class origin and terrorist recruitment. 17.The method of claim 1, wherein creating a neighborhood includes mappinga grid onto a sphere, thereby providing eight neighboring cells for eachcell.
 18. The method of claim 1, wherein iteratively conducting discreteinteractions includes one agent interacting with multiple agents in asingle iteration.
 19. The method of claim 1, wherein iterativelyconducting discrete interactions includes an agent interacting with anon-neighboring agent.
 20. The method of claim 1, wherein recordinginteraction outcome includes recording coalition dynamics, including atleast one of coalitions, coalition size, join count, total resources,and coalition locations.
 21. The method of claim 1, further comprisingcollecting a predetermined measure of resources from each agent afterconducting each discrete interaction between the agents.
 22. The methodof claim 1, further comprising displaying Fourier transform of at leastone of coalition size and join count.
 23. A non-transitory,computer-readable medium having machine-executable code for simulatingthe interactions of agents, comprising: code for creating a neighborhoodof cells; code for allocating resources to the cells; code for assigningan agent to each cell; code for selecting one of a plurality of decisionrules for interactions between agents, wherein one of the plurality ofdecision rules includes a coordination game; code for analyzing risksensitivity using an analytical function W(x)=e^((d+ax+c sin(bx))),where a>bc; code for iteratively conducting discrete interactionsbetween the agents using the selected decision rule and allocatedresources; and code for recording interaction outcome.
 24. Thenon-transitory, computer-readable medium of claim 23, wherein thecoordination game includes a mixed strategy matrix of: Join Defect JoinR, R S, T Defect T, S P, P

wherein payoffs include R>P=T>S.
 25. The non-transitory,computer-readable medium of claim 23, wherein one of the plurality ofdecision rules includes a conformist strategy.
 26. The non-transitory,computer-readable medium of claim 23, wherein one of the plurality ofdecision rules includes a prestige-bias strategy.
 27. Thenon-transitory, computer-readable medium of claim 23, wherein one of theplurality of decision rules includes a modified risk posture based on atleast one of a sigmoid and an expo-sigmoid function for distribution ofresources among agents.
 28. The non-transitory, computer-readable mediumof claim 23, wherein allocating resources includes allocating resourcesaccording to a function of a data set, the function including at leastone of a sigmoid and an expo-sigmoidal function derived from Fouriersignature analysis fit of residuals from at least one of a linearregression to the data set and a logarithm of the data set.
 29. Thenon-transitory, computer-readable medium of claim 23, wherein the one ofthe plurality of decision rules includes a join probability according toa function of a data set, the function including at least one of asigmoid and an expo-sigmoidal function derived from Fourier signatureanalysis fit of residuals from at least one of a linear regression tothe data set and a logarithm of the data set.
 30. The non-transitory,computer-readable medium of claim 23, wherein at least one of theresources and the one of a plurality of decision rules includes afunction of a data set, the data set including a measure of at least oneof well-being, social status, and social worth.
 31. The non-transitory,computer-readable medium of claim 23, further comprising code foranalyzing recorded interaction outcome.
 32. The non-transitory,computer-readable medium of claim 31, wherein the analyzing recordedinteraction outcome includes determining an effect on risk sensitivityof the measure of at least one of well-being, social status, and socialworth.
 33. The non-transitory, computer-readable medium of claim 31,wherein analyzing recorded interaction outcome includes determining atleast one of an individual's likelihood of taking a risk, a distributionof well-being, a distribution of social status, a distribution of socialworth, a potential for social unrest due to wealth inequalities, animpact of nepotism on social unrest, an influence of class origin ofinsurgency members, and an influence of class origin and terroristrecruitment.
 34. The non-transitory, computer-readable medium of claim23, wherein creating a neighborhood includes mapping a grid onto asphere, thereby providing eight neighboring cells for each cell.
 35. Thenon-transitory, computer-readable medium of claim 23, whereiniteratively conducting discrete interactions includes one agentinteracting with multiple agents in a single iteration.
 36. Thenon-transitory, computer-readable medium of claim 23, whereiniteratively conducting discrete interactions includes an agentinteracting with a non-neighboring agent.
 37. The non-transitory,computer-readable medium of claim 23, wherein recording interactionoutcome includes recording coalition dynamics, including at least one ofcoalitions, coalition size, join count, total resources, and coalitionlocations.
 38. The non-transitory, computer-readable medium of claim 23,further comprising code for collecting a predetermined measure ofresources from each agent after conducting each discrete interactionbetween the agents.